Motion of a Point

Position, Velocity, and Acceleration

• Average Velocity

• Instantaneous Velocity

t

rv

rdt

rdv

Acceleration

• Average Acceleration

• Instantaneous Acceleration

t

va

rdt

rd

dt

vda

2

2

Uniform Rectilinear Motion

tvss

tvxx

vv

00

00

0

Uniformly Accelerated Rectilinear Motion

)(2

2

1

020

2

200

0

xxavv

tatvxx

tavv

Other Equations

Acceleration

dx

dvva

Acceleration as a function of velocity

v

v

t

t

dtva

dv

dtva

dv

vadt

dv

0 0)(

)(

)(

v

v

s

s

dsva

vdv

dsva

vdv

vavds

dv

vdx

dv

dt

dx

dx

dv

dt

dvva

0 0)(

)(

)(

)(

Acceleration as a function of Position

s

s

v

v o

dssavdv

dssavdv

savds

dv

vds

dv

dt

ds

ds

dv

dt

dv

sadt

dv

)(

)(

)(

)(

0

The last integral yields velocity as a function of position.

s

s

t

t

dtsv

ds

dt

dssv

0 0)(

)(

Erratic Motion

• Motion described by a piecewise function.

• Graph of s, v, a, and t.

• Motion Sensor

Position vs. Time

• Slope is velocity

• Second derivative is acceleration

Velocity vs. Time

• Slope is acceleration

• Area is displacement

Acceleration vs. Time

• Area is change in velocity.

Acceleration vs. Position

1

0

20

212

1s

s

adsvv

Velocity vs. Position

ds

dvva

Curvilinear Motion of Particles

Position Vector, Velocity, and Acceleration

rvdt

vda

rdt

rdv

Derivatives of Vector Functions

• Summation Rule

• Product Rule– Dot Product– Cross Product

Rectangular Components of Velocity and Acceleration

• Basically a summation rule.

Angular Motion

dt

ddt

d

Tangential and Normal Components

• Coordinate system

Unit vector

tt

tn

evedt

dsv

d

ede

ˆˆ

ˆˆ

nt

tt

tt

ev

edt

dv

dt

ds

ds

d

d

edve

dt

dvdt

edve

dt

dv

dt

vda

ˆˆ

ˆˆ

ˆˆ

2

2

2

2

32

1

)(

ˆcosˆsinˆ

ˆsinˆcosˆ

dxyd

dxdy

xyy

jie

jie

n

t

At a given instant in an airplane race, airplane A is flying horizontally in a straight line, and its speed is being increased at a rate of 6 m/s2. Airplane B is flying at the same altitude as airplane A and, as it rounds a pylon, is following a circular path of 2000-m radius. Knowing that at the given instant the speed of B is being decreased at the rate of 2 m/s2 determine, for the positions shown, (a) the velocity of B relative to A, (b) the acceleration of B relative to A.

A car travels at 100 km/h on a straight road of increasing grade whose vertical profile can be approximated by the equation shown. When the car’s horizontal coordinate is x = 400 m, what are the tangential and normal components of the car’s acceleration?

Polar and Cylindrical – Radial and Transverse

errerra

ererv

r

r

ˆ2ˆ

ˆˆ

2

1 and , ,

1

t

CtzBt

t

Ar

The three-dimensional motion of a particle is defined by the cylindrical coordinates

Determine the magnitudes of the velocity and acceleration when(a) t = 0 and (b) t = infinity

Dependent Motion

• Measure positions with respect to a fixed point.

• There will generally be a physical constraint, often a rope or cable.

At the instant shown, slider block B is moving to the right with a constant acceleration, and its speed is 6 in./s. Knowing that after slider block A has moved 10 in. to the right its velocity is 2.4 in./s, determine the accelerations of A and B.

Slider block B starts from rest and moves to the right with a constant acceleration of 1 ft/s/s. Determine the relative acceleration of portion C of the cable with respect to slider block A.

Relative Motion

• Notation – the position of B relative to A

ABr

Racing cars A and B are traveling on circular portions of a race track. At the instant shown, the speed of A is decreasing at the rate of 8 m/s2 and the speed of B is increasing at the rate of 3 m/s2 . For the positions shown, determine

a) the velocity of B relative to Ab) the acceleration of B relative to A.