Introduction to Motion in a Plane

The physical quantities like work, temperature and distance can be

represented in day to day life wholly by their magnitude alone.

However, the relation of these physical quantities can be explained by

the laws of arithmetic. In order to represent physical quantities like

acceleration, displacement, and force, the direction is equally essential

along with the magnitude. Let us now study Plane Motion.

Introduction to Plane Motion

Velocity refers to a physical vector quantity which is described by

both magnitude and direction. The magnitude or scalar absolute value

of velocity is referred to as speed. As stated by the Pythagorean

Theorem, the magnitude of the velocity vector is given by –

| v | = v = √ ( vx ²+ vy ² )

Acceleration is defined by the rate of change of velocity of an object

with respect to time. Numerically or in terms of components, it can be

presented as –

ax =

d

dt

vx

ay =

d

dt

vy

Motion in a Plane

Motion in a plane is also referred to as a motion in two dimensions.

For example, circular motion, projectile motion, etc. For the analysis

of such type of motion, the reference point will be made of an origin

and the two coordinate axes X and Y.

Motion in a plane refers to the point where we consider motion in two

dimensions as only two dimensions makes a plane. Here, considering

the above, we take two axes into consideration – generally X-axis or Y

– axes. In an attempt to derive the equation of the motion in a plane,

we must know about motion in one direction.

Equations of Plane Motion

The equations of motion in a straight line are:

v = u+at

s = ut+1/2 at²

v2 = u² + 2as

Where,

● v = final velocity of the particle

● u = initial velocity of the particle

● s = displacement of the particle

● a = acceleration of the particle

● t = the time interval in which the particle is in consideration

In a plane, we have to apply the same equations separately in both the

directions: Y axis and Y-axis. This would give us the equations for

motion in a plane.

vy = uy + ayt

sy = uy t +1/2 ay t²

v²y = u²y+2ay s

Where,

● vy = final velocity of the particle in the y-direction

● uy = initial velocity of the particle in the y-direction

● sy = displacement of the particle in the y-direction

● ay = acceleration of the particle in the y-direction

Similarly, for the X-axis :

vx = ux + ax t

sx= ux t+1/2 axt²

v²x = u²x+2 axs

Where,

● vx = final velocity of the particle in the x-direction

● ux = initial velocity of the particle in the x-direction

● sx= displacement of the particle in the x-direction

● ax = acceleration of the particle in x-direction

Projectile Motion: Plane Motion

One of the most common examples of motion in a plane is Projectile

motion. In a projectile motion, the only acceleration acting is in the

vertical direction which is acceleration due to gravity (g). Therefore,

equations of motion can be applied separately in the X-axis and Y-axis

to find the unknown parameters.

The above diagram represents the motion of an object under the

influence of gravity. It is an example of projectile motion (an special

case of motion in a plane).

Examples of Two-Dimensional Plane Motion

● Throwing a ball or a cannonball

● The motion of a billiard ball on the billiard table.

● A motion of a shell fired from a gun.

● A motion of a boat in a river.

● The motion of the earth around the sun.

Solved Examples for You

Question 1. The state with reasons, whether the following algebraic

operations with scalar and vector physical quantities are meaningful:

a. adding any two scalars,

b. adding a scalar to a vector of the same dimensions,

c. multiplying any vector by any scalar,

d. multiplying any two scalars,

e. adding any two vectors,

f. adding a component of a vector to the same vector.

Solution.

a. Meaningful; The addition of two scalar quantities is meaningful

only if they both represent the same physical quantity.

b. Not Meaningful; The addition of a vector quantity with a scalar

quantity is not meaningful.

c. Meaningful; A scalar can be multiplied with a vector. For

example, force is multiplied with time to give impulse.

d. It’s meaningful; A scalar, irrespective of the physical quantity

it represents, can be multiplied by another scalar having the

same or different dimensions.

e. Meaningful; The addition of two vector quantities is

meaningful only if they both represent the same physical

quantity.

f. Meaningful; A component of a vector can be added to the same

vector as they both have the same dimensions.

Question 2: Read each statement below carefully and state with

reasons, if it is true or false:

(a) The magnitude of a vector is always a scalar,

(b) each component of a vector is always a scalar,

Solution :

(a) True. The magnitude of a vector is a number. Hence, it is a scalar.

(b) False. Each component of a vector is also a vector.

Scalars and Vectors

In today’s world, various mathematical quantities depict the motion of

objects into two categories. The quantity is as either a vector or a

scalar quantity which distinguishes from one another by their

difference and distinct definitions. Let’s study more about the scalars

and vectors below.

Scalars and Vector Quantities

● Scalar Quantities: The physical quantities which are specified

with the magnitude or size alone are scalar quantities. For

example, length, speed, work, mass, density, etc.

● Vector Quantities: Vector quantities refer to the physical

quantities characterized by the presence of both magnitude as

well as direction. For example, displacement, force, torque,

momentum, acceleration, velocity, etc.

Comparison between Scalars and Vectors

Criteria Scalar Vector

Definition A scalar is a quantity with magnitude only.

A vector is a quantity with the magnitude as well as direction.

Direction No direction Yes there is the direction

Specified by A number (Magnitude) and a Unit

A number (magnitude), direction and a unit.

Represented by Quantity symbol Quantity symbol in bold or an arrow

sign above

Example Mass and Temperature Velocity and Acceleration

Characteristics of Vectors

The characteristics of vectors are as followed –

● They possess both magnitudes as well as direction.

● They do not obey the ordinary laws of Algebra.

● These change if either the magnitude or direction change or

both change.

Browse more Topics under Motion In A Plane

● Introduction to Motion in a Plane

● Resolution of Vectors and Vector Addition

● Addition and Subtraction of Vectors – Graphical Method

● Relative Velocity in Two Dimensions

● Uniform Circular Motion

● Projectile Motion

Unit Vector

A unit vector is that vector which is a vector of unit magnitude and points in a particular direction. The unit vector in the direction of

A

is

A

^

and is defined by –

| A |

A

^

=

A

The unit vectors along the x, y and z-axis is

i

^

,

j

^

, and

k

^

respectively.

Equal Vectors

Vectors A and B are equal if | A | = | B | as well as their directions, are

same.

Zero Vectors

Zero vector is a vector with zero magnitudes and an arbitrary direction

is a zero vector. It can be represented by O and is a Null Vector.

Negative of a Vector

The vector whose magnitude is same as that of a (vector) but the

direction is opposite to that of a ( vector ) is referred to as the negative

of a ( vector ) and is written as – a ( vector ).

Parallel Vectors

A and B are said to be parallel vectors if they have the same direction,

or may or may not have equal magnitude ( A || B ). If the directions

are opposite, then A ( vector ) is anti-parallel to B ( vector ).

Coplanar Vectors

If the vectors lie in the same plane or they are parallel to the same

plane, the vectors are said to be coplanar. If not, the vectors are said to

be non – planar vectors.

Displacement Vectors

The displacement vector refers to that vector which gives the position

of a point with reference to a point other than the origin of the

coordinate system.

Solved Question for You

Question 1: State for each of the following physical quantities, if it is a

scalar or a vector.

Volume, Mass, Speed, Velocity, Displacement, Acceleration, Density,

Number of Moles, Angular Frequency, Angular Velocity,

Displacement

Solution.

● A scalar is the one that is specified by its magnitude. It does not

have any direction associated with it. Some of the scalar

physical quantities are – Volume, Speed, Mass, Density,

Number of Moles and Angular Frequency.

● Whereas a Vector quantity is the one which is specified by its

magnitude as well as its direction that is associated with it.

Some of the vector quantity are velocity, acceleration,

displacement and angular velocity.

Therefore,

● Scalar: Volume, Mass, Speed, Velocity, Density, Number of

Moles, Angular Frequency

● Vector: Acceleration, Velocity, Displacement, Angular

Velocity.

Question 2: Pick out the two scalar quantities in the list.

Force, Work, Angular Momentum, Current, Linear Momentum,

Electric field, Average Velocity, Relative Velocity, Magnetic Moment

Solution. The scalar quantities from the mentioned are Work and

Current.

● Work done is the dot product of force and displacement. Since

the dot product of two quantities is always a scalar, therefore,

work is a scalar physical quantity.

● Current refers to the one in which the direction is not taken into

consideration and is described only by its magnitude.

Henceforth, it is a scalar quantity.

Resolution of Vectors and Vector Addition

In contrast to the concept of addition, the concept of resolution of vectors could be well understood. For say, in the process of addition, let’s consider

a

and

b

are directly added to get

S

. Now, let

S

be broken down to obtain

a

and

b

back. This is referred to as the process of resolution. Let us study the

chapter resolution of vectors and vector addition in detail.

Resolution of Vectors

A vector can be resolved into many different vectors, for resolution of

vectors. For Example: Let us consider two numbers, say, 4 and 6,

which is further added to obtain 10. Further, now 10 is broken or

resolved. However, the number 10 can also be resolved into many

other numbers like –10 = 5 + 5; 10 = 3 + 7 etc.

In a similar way, a vector can essentially be further broken or resolved

to obtain multiple vectors of different magnitudes and directions. In

Physics, vectors would be mainly resolved only along the coordinate

axes, X, Y, and Z.

Browse more Topics under Motion In A Plane

● Introduction to Motion in a Plane

● Scalars and Vectors

● Addition and Subtraction of Vectors – Graphical Method

● Relative Velocity in Two Dimensions

● Uniform Circular Motion

● Projectile Motion

Resolving Vectors along X and Y Axis

To resolve a vector on an X-Y plane, first, draw the vector. Then,

label and create the constructions on the figure as mentioned below :

The complete figure seems to be like a parallelogram, further applying the Parallelogram law of vector addition. Herein, the two vectors

a

x

and

a

y

appears to be added by the parallelogram law of vector addition to obtain

a

. Therefore, with this, we can say that

a

x

and

a

y

are the resolved output of

a

as

a

has been again broken back to its components. Here,

● a ● ● x

● is the x-component; and

● a ● ● y

● is the y-component of ● a ●

● .

In order to find the magnitudes of each component, in △OBC:

cos θ = OBOC

Therefore, OB = OC cos θ

Hence, |

a

x

| = |

a

| cos θ [ Magnitude of x – component ]

sin θ = BCOC

So, BC = OC sin θ;

Also, BC = OD ( Opposite sides of Rectangle );

Therefore, |

a

y

| = |

a

| sin θ [ Magnitude of y – component ]

Note

The General Rule of Thumb states that the subtended angle will

always touch one of the components. Further, the component which

the given angle touches or the given angle is subtended with will be

the cos component of the given vector. Whereas the other will

automatically be the sin component.

Unit Vectors along Co-ordinate Axes

For the co-ordinate axes, there are special unit vectors designated by

convention î, ĵ, and k, which respectively represents X, Y, and Z axes.

By the property of unit vectors, |î| = |ĵ| = |k| = 1

Writing Vectors in Component Form: Using Coordinate Axes Unit Vectors

A vector can be expressed as a product of its magnitude and direction.

Therefore,

● a ●

● x

● = | ● a ● ● | cos θ î. The magnitude of ● a ● ● x

● , which is | ● a ● ● x

● |, is multiplied to the direction in which ● a ● ● x

● lies, which is the direction of x-axis, represented by the unit

vector î.

● a ● ● y

● = | ● a ● ● | sin θ ĵ. The magnitude of ● a ● ● y

● , which is |

● a ● ● y

● |, is multiplied to the direction in which| ● a ● ● y

● | lies, which is the direction of y-axis, represented by the unit

vector ĵ.

● Now we know that ● a ● ● = ● a ● ● x

● + ● a ● ● y

● .

Therefore,

a

= (|

a

| cos θ ) î + (|

a

| sin θ ) ĵ.

Addition of Vectors and Subtraction of Vectors in Component Form (Expressed as Unit Vectors)

Consider the following figure:

Expressing

a

in component form,

● a ● ● = ● a ● ● x

● + ● a ● ● y

● ;

● a ● ● = | ● a ● ● x

● | î + | ● a ● ● y

● | ĵ;

● a ● ● = (| ● a ● ● | cos θ ) î + (| ● a ● ● | sin θ ) ĵ

Expressing

b

in component form,

● b ● ● = ● b ● ● x

● + ● b ● ● y

● ;

● b ● ● = | ● b ● ● x

● | î + | ● b ● ● y

● | ĵ;

● b ●

● = (| ● b ● ● | cos α ) î + (| ● b ● ● | sin α ) ĵ

Operation, (\vec{S}\) =

a

+

b

;

● S ● ● = ● a ●

● + ● b ● ● = (|ax| î + |a y| ĵ) + (|b x| î + |b y| ĵ);

● This implies, ● S ● ● = (|ax| + |bx|) î + (|a y| + |by|) ĵ;

● Hence, ● S ● ● = (|a| cos θ + |b| cos α) î + (|a| sin θ + |b| sin α) ĵ [Final Sum]

Operation,

S

=

a

–

b

;

● S ● ● = ● a ● ● – ● b ● ● = (|ax| î – |a y| ĵ) + (|b x| î – |b y| ĵ);

● This implies, ● S ● ● = (|ax| – |bx|) î + (|a y| – |by|) ĵ;

● Hence, ● S ●

● = (|a| cos θ – |b| cos α) î + (|a| sin θ – |b| sin α) ĵ [Final

Difference]

Solved Examples for You

Example: A laser beam is aimed 15.95° above the horizontal at a

mirror 11,648 m away. It glances off the mirror and continues for an

additional 8570. m at 11.44° above the horizon until it hits its target.

What is the resultant displacement of the beam to the target?

Solution: Let us break up the vector into their components –

x1 = r1 cos θ1

x1 = (11,648 m)cos(15.95°)

x1 = 11,200 m

y1 = r1 sin θ1

y1 = (11,648 m)sin(15.95°)

y1 = 3,200 m

x2 = r2 cos θ2

x2 = (8,570 m)cos(11.44°)

x2 = 8,400 m

y2 = r2 sin θ2

y2 = (8,570 m)sin(11.44°)

y2 = 1,700 m

Add vectors in the same direction with “ordinary” addition. –

x =

11,200 m + 8,400 m

x = 19,600 m

y = 3,200 m + 1,700 m

y = 4,900 m

Add vectors at right angles with a combination of pythagorean

theorem for magnitude…

r = √(x2 + y2)

r = √[(19,600 m)2 + (4,900 m)2]

r = 20,200 m

and tangent for direction.

tan θ =

y

=

4,900 m

x 19,600 m

θ = 14.04°

Addition and Subtraction of Vectors

In Physics, vector quantities are quantities that have a magnitude and

direction. It is important to understand how operations like addition

and subtraction are carried out on vectors. In this chapter, we will

learn about these quantities and their addition and subtraction

operations. Let us begin with the addition of vectors.

Scalar and Vector Quantities

● Scalar Quantities: The physical quantities which are specified

with the magnitude or size alone are referred to as Scalar

Quantities. It is one – dimensional measurement of quantity

like mass or temperature. A physical quantity which has force

but no direction. For example, length, speed, work, mass,

density, etc.

● Vector Quantities: Vector Quantities refers to those physical

quantities which are characterized by the presence of both

magnitudes as well as direction. It is indeed a quantity that

requires both of magnitude and direction in order to identify the

right quantity. For example, displacement, force, torque,

momentum, acceleration, velocity, etc.

Browse more Topics under Motion In A Plane

● Introduction to Motion in a Plane

● Scalars and Vectors

● Resolution of Vectors and Vector Addition

● Relative Velocity in Two Dimensions

● Uniform Circular Motion

● Projectile Motion

Characteristics of Vectors

The characteristics of vectors are as follows

● They possess both magnitudes as well as direction.

● They do not obey the ordinary laws of Algebra.

● These change if either the magnitude or direction change or

both change.

● Vectors are significantly represented by the bold-faced letters

or letter which have an arrow over them.

Types of Vectors

Unit Vector

A unit vector is that vector which is a vector of unit magnitude and points in a particular direction. It is specifically used for the direction only. The unit vector is represented by putting a cap (^) over the specified quantity. The unit vector in the direction of

A

is denoted by

A

^

and is defined by,

| A |

A

^

=

A

The unit vectors along the x, y and z-axis can be denoted as

i

^

,

j

^

, and

k

^

respectively.

Equal Vectors

Vectors A and B are said to be equal if |A| = |B| as well as their

directions, are same.

Zero Vectors

Zero vector is a vector with zero magnitudes and an arbitrary direction

is called a zero vector. It can be represented by O and is also known as

Null Vector.

Negative of a Vector

The vector whose magnitude is same as that of a (vector) but the

direction is opposite to that of a (vector) is referred to as the negative

of a (vector) and is written as – a (vector).

Addition of Vectors

As per the geometrical method for the addition of vectors, two vectors

a and b. By drawing them to a common scale and placing them

according to head to tail, it may be added geometrically. The vector

that gets connected to the tail of the first to the head of the second is

the sum of vector c. The vector addition obeys the law of associativity

and is commutative.

Analytical Method i.e. Parallelogram Law for Addition of Vectors

If the two vector a and b are given such that the angle between them is

θ, in that case, the magnitude of the resultant vector c of the addition

of vectors is stated by –

c = √ (a2 + b2 + 2abcos θ)

And its direction is given by an angle Φ with vector a,

Tan Φ = b sin θ / (a + b cos θ)

Subtraction of Vectors

In order to subtract vector b from a, the direction must be reverse of

vector b to get vector (-b). Then it must be added : (-b) to a.

Vectors Addition Vectors Subtraction of Vectors

A = Ax î +Ay ĵ

and

B = Bx î

+By ĵ

R = A + B

R = Rx î + Ry ĵ

where

Rx = Ax + Bx

and

Ry = Ay + By

R = A – B

R = Rx î – Ry ĵ

where

Rx = Ax – Bx

and

Ry = Ay – By

A = Ax î +Ay ĵ+Az k

and

B = Bx î

+By ĵ+Bz

k

R = A + B

R = Rxî + Ryĵ + Rzk

where

Rx = Ax + Bx

and

Ry = Ay + By

and

Rz = Az – Bz

R = A – B

R = Rxî + Ry ĵ + Rz k

where

Rx = Ax – Bx

and

Ry = Ay – By

and

Rz = Az – Bz

Solution Examples for You

Question: Read each statement below carefully and state, with reasons

and examples, if it is true or false :

A scalar quantity is one that –

1. is conserved in a process

2. can never take negative values

3. has the same value for observers with different orientations of

axes

4. must be dimensionless

5. does not vary from one point to another in space

Solution:

1. False: Despite being a scalar quantity, energy is not conserved

in inelastic collisions.

2. False: Despite being a scalar quantity, the temperature can take

negative values.

3. True: The value of a scalar does not vary for observers with

different orientations of axes.

4. False: Total path length is a scalar quantity. Yet it has the

dimension of length.

5. False: A scalar quantity like that of a gravitational potential can

vary from one point to another in space.

Question: If a = 2i + j and b = 4i + 7j

1. Find the components of vector c = a + b

2. Find the magnitude of c and its angle with x – axis.

Solution:

1. c = a + b = (2i + j) + (4i + 7j) ⇒ C = (2 + 4) i + (1 + 7) j ⇒

Thus, cx = 6 and cy = 8

2. c = √ cx2 + cy2 = √ (6)2 + (8)2 = 100 ⇒ tan θ = cy/cx = 8/6 =

4/3 ⇒ θ = tan -1 (4/3) = 53 degree

Relative Velocity in Two Dimensions

The concept of relative motion velocity in a plane is quite similar to

the whole concept of relative velocity in a straight line. Considering

various occasions, we take in more than one object move in a frame

which is non-stationary in respect to another viewer.

What is Relative Motion Velocity?

The relative motion velocity refers to an object which is relative to

some other object that might be stationary, moving with the same

velocity, or moving slowly, moving with higher velocity or moving in

the opposite direction. The wide concept of relative velocity can be

very easily extended for motion along a straight line, to include

motion in a plane or either in three dimensions.

Velocity

Velocity refers to a physical vector quantity which is described by

both magnitude and direction. The magnitude or scalar absolute value

of velocity is referred to as speed. As stated by the Pythagorean

Theorem, the magnitude of the velocity vector is given by –

| v | = v = √ ( vx ²+ vy ² )

Browse more Topics under Motion In A Plane

● Introduction to Motion in a Plane

● Scalars and Vectors

● Resolution of Vectors and Vector Addition

● Addition and Subtraction of Vectors – Graphical Method

● Uniform Circular Motion

● Projectile Motion

Acceleration

Acceleration is the rate of change of velocity of an object with respect

to time. Numerically or in terms of components, it can be presented as

–

ax =

d

dt

vx

ay =

d

dt

vy

Relative Motion Velocity in Two Dimensions

Let us consider two objects A and B who are moving with the velocity

at Va and Vb with respect to some common frame of reference, for

example, may be the ground or the earth. In this case, the velocity of

object A relative to that of B will be –

Vab= Va – Vb

Similarly, the velocity of object B relative to that of A will be :

Vba = Vb– Va

Therefore,

Vba= – Vba

And | Vab | = | Vba |

When two objects seem to be stationary for one another, in that case –

Vb = Va

Vba = Vab = 0

The magnitude of Vba and Vab will be lower than the magnitude of Va

and Vb if Va and Vb are of same sign. Object A appears faster to B and

B appears slower to A if Va > Vb.

The magnitude of Vba and Vab will be higher than the magnitude of Va

and Vb if Va and Vb are of opposite sign. In this case, both objects will

appear moving faster to one another.

Solved Questions for You

Q1: If rain is falling vertically at a speed of 35 m/s and a person is

riding a bicycle at 12 m/s ( east to west ) then the relative motion

velocity of rain will be Vrb

Solution : Vrb = Vr – Vb = ( 35 – 12 ) m/s = 23 m/s

Tan θ = Vw / Vr = 12 / 35 = 0.343

θ = 19 degree

The person must hold an umbrella at an angle of 19 degrees with

vertical towards the west to avoid the rain.

Q2. A plane is traveling at velocity 100 km/hr, in the southward

direction. It encounters wind traveling in the west direction at a rate of

25 km/hr. Calculate the resultant velocity of the plane.

Solution: Given, the velocity of the wind = Vw = 25km / hr

The velocity of the plane = Va = 100 km / hr

The relative motion velocity of the plane with respect to the ground

can be given as the angle between the velocity of the wind and that of

the plane is 90°. Using the Pythagorean theorem, the resultant velocity

can be calculated as,

R2= (100 km/hr)² + (25 km/hr)²

R2 = 10 000 km² / hr² + 625 km² / hr²

= 10 625 km² / hr²

Hence, R = 103.077 km/hr

Using trigonometry, the angle made by the resultant velocity with

respect to the horizontal plane can be given as,

Tan θ = ( window velocity / airplane velocity )

Tan θ =( 25/100 )

θ = tan −1(1/4)

θ =1/4 radians

Uniform Circular Motion

The term “Uniform circular motion” is the kind of motion of an object

in a circle at a constant speed. With uniform circular motion, as an

object moves in a particular circle, the direction of it is constantly

changing. Herein, in all instances, the object is moving tangent to the

circle. Let us study this angular motion in detail.

Circular Motion

Circular motion is the movement of an object in a circular path.

Uniform Circular Motion

This motion refers to the circular motion if the magnitude of the

velocity of the particle in circular motion remains constant. The

non-uniform circular motion refers to the circular motion when the

magnitude of the velocity of the object is not constant. Another special

kind of circular motion is when an object rotates around itself also

known by spinning motion.

Variables in Circular Motion

Angular Displacement

The angle which is subtended by the position vector at the center of

the circular path refers to the angular displacement.

Angular Displacement (Δθ) = (ΔS/r)

Where Δ’s refers to the linear displacement while r is the radius.

Radian is the unit of Angular Displacement.

Browse more Topics Under Motion In A Plane

● Introduction to Motion in a Plane

● Scalars and Vectors

● Resolution of Vectors and Vector Addition

● Addition and Subtraction of Vectors – Graphical Method

● Relative Velocity in Two Dimensions

● Uniform Circular Motion

● Projectile Motion

Angular Acceleration

It refers to the rate of time of change of angular velocity (dῶ).

Angular acceleration (α) = dῶ/dt = d2θ / dt2

Its unit is rad/s2 and dimensional formula [T]-2. The relation between

linear acceleration (a) and angular acceleration (α)

A = rα, where r is the radius.

Angular Velocity

It refers to the time rate change of angular displacement (dῶ).

Angular Velocity (ῶ) = Δθ/Δt

Angular Velocity is a vector quantity. Its unit is rad/s. The relation

between the linear velocity (v) and angular velocity (ῶ) is

v = rῶ

Centripetal Acceleration

It refers to an acceleration that acts on the body in circular motion

whose direction is always towards the center of the path.

Centripetal Acceleration (α) = v2/r = rῶ2.

The magnitude of this acceleration by comparing ratios of velocity

and position around the circle. Since the particle is traveling in a

circular path, the ratio of the change in velocity to velocity will be the

same as the ratio of the change in position to position. It is also known

as radial acceleration as it acts along the radius of the circle.

Centripetal Acceleration is a vector quantity and the unit is in m/s2.

Solved Question for You

Q. During the course of a turn, an automobile doubles its speed. How

much must additional frictional force the tires provide if the car safely

makes around the curve? Since Fc varies with v2, an increase in

velocity by a factor of two must be accompanied by an increase in

centripetal force by a factor of four.

A satellite is said to be in geosynchronous orbit if it rotates around the

earth once every day. For the earth, all satellites in geosynchronous

orbit must rotate at a distance of 4.23×107 meters from the earth’s

center. What is the magnitude of the acceleration felt by a

geosynchronous satellite?

Solution: The acceleration felt by any object in uniform circular

motion is given by

a =

v²÷r

We are given the radius but must find the velocity of the satellite. We

know that in one day, or 86400 seconds, the satellite travels around

the earth once. Thus:

v =

Δr÷Δt

=

2πr÷Δt

=

2π×4.23×

10

7

÷86400

= 3076m/s

a =

v

2

÷r

=

3076

2

÷(4.23×

10

7

)

= .224m/s²

Projectile Motion

Motion in a plane is also referred to as a motion in two dimensions.

For example – Circular Motion, Projectile Motion, etc. For the

analysis of such type of motion (i.e. Projectile Motion), the reference

point will be made of an origin and the two coordinate axes X and Y.

One of the most common examples of motion in a plane is Projectile

motion.

Projectile Motion

Projectile refers to an object that is in flight after being thrown or

projected. In a projectile motion, the only acceleration acting is in the

vertical direction which is acceleration due to gravity (g). Equations of

motion, therefore, can be applied separately in X-axis and Y-axis to

find the unknown parameters.

Some examples of Projectile Motion are Football, A baseball, A

cricket ball, or any other object. The projectile motion consists of two

parts – one is the horizontal motion of no acceleration and the other

vertical motion of constant acceleration due to gravity. The projectile

motion is always in the form of a parabola which is represented as:

Browse more Topics under Motion In A Plane

● Introduction to Motion in a Plane

● Scalars and Vectors

● Resolution of Vectors and Vector Addition

● Addition and Subtraction of Vectors – Graphical Method

● Relative Velocity in Two Dimensions

● Uniform Circular Motion

y = ax + bx2

Projectile motion is calculated by a way of neglecting air resistance in

order to simplify the calculations. The above diagram represents the

motion of an object under the influence of gravity. It is an example of

projectile motion (a special case of motion in a plane). The motion of

a projectile is considered as a result:

Few Examples of Two – Dimensional Projectiles

● Throwing a ball or a cannonball

● The motion of a billiard ball on the billiard table.

● A motion of a shell fired from a gun.

● A motion of a boat in a river.

● The motion of the earth around the sun.

What is Motion in Plane?

When a particle is projected obliquely near the earth’s surface, it

moves simultaneously in the direction of horizontal and vertical. The

motion of such a particle is called Projectile Motion. In the above

diagram, where a particle is projected at an angle θ, with an initial

velocity u. For this particular case, we will calculate the following:

● The time is taken to reach point A from O

● The horizontal distance covered (OA)

● The maximum height reached during the motion.

● The velocity at any time “ t “ during the motion.

Quantity Value

Components of velocity at time t vx = v0 cosθ0

vy = v0 sinθ0–gt

Position at time t

x = (v0 cosθ0)t

y = (v0 sinθ0)t – 1/2 gt2

Equation of path of projectile motion y = (tan θ0)x – gx2/2(v0cosθ0)2

Time of maximum height tm = v0 sinθ0 /g

Time of flight 2tm = 2(v0 sinθ0/g)

Maximum height of projectile hm = (v0 sinθ0)2/2g

Horizontal range of projectile R = v02 sin 2θ0/g

Maximum horizontal range ( θ0 = 45° ) Rm = v02/g

If any object is thrown with the velocity u, making an angle Θ from

horizontal, then the horizontal component of initial velocity = u cos Θ

and the vertical component of initial velocity = u sin Θ. The horizontal

component of velocity (u cos Θ) remains the same during the whole

journey as herein, no acceleration is acting horizontally.

The vertical component of velocity (u sin Θ) gradually decrease and at

the highest point of the path becomes 0. The velocity of the body at

the highest point is u cos Θ in the horizontal direction. However, the

angle between the velocity and acceleration is 90 degree.

Important Points of Projectile Motion

● The linear momentum at the highest point is mu cos Θ and the

kinetic energy is (1/2)m(u cos Θ)2

● After t seconds, the horizontal displacement of the projectile is

x = (u cos Θ) t

● After t seconds, the vertical displacement of the projectile is y

= (u sin Θ) t – (1/2) gt2

● The equation of the path of the projectile is y = x tan Θ –

[g/(2(u2 cos Θ)2)]x2

● The path of a projectile is parabolic.

● At the lowest point, the kinetic energy is (1/2) mu2

● At the lowest point, the linear momentum is = mu

● Throughout the motion, the acceleration of projectile is

constant and acts vertically downwards being equal to g.

● The angular momentum of projectile = mu cos Θ × h where the

value of h denotes the height.

● The angle between the velocity and acceleration in the case of

angular projection varies from 0 < Θ < 180 degrees.

What is Relative Velocity in Two Dimensions?

Solved Examples for You

Question: A man can swim at a speed of 4.0 km/h in still water. How

long does he take to cross a river 1.0 km wide if the river flows

steadily at 3.0 km/h and he makes his strokes normal to the river

current? How far down the river does he go when he reaches the other

bank?

Solution: Speed of the man = 4 km/h and width of the river = 1 km

Time taken to cross the river = Width of the river / Speed of the river

= 1/4 h = 1/4 × 60 = 15 minutes

Speed of the river, vr = 3 km/h

Distance covered with flow of the river = vr × t

= 3 × 1/3 = 1/4

= 3/4 × 1000 = 750 m