Physics 215 – Fall 2016 Slide 02-2 1

Welcome back to

Physics 215

Lecture 2-2

Physics 215 – Fall 2016 Slide 02-2 2

•  Last time: – Displacement, velocity, graphs

•  Today: – Constant acceleration, free fall

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2-2.1: An object moves with constant acceleration, starting from rest at t = 0 s. In the first four seconds, it travels 10 cm. What will be the displacement of the object in the following four seconds (i.e. between t = 4 s and t = 8 s)?

A.  10 cm B. 20 cm C.  30 cm D.  40 cm

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Motion with constant acceleration:

v = vi + at

vav = (1/2) (vi + v)

x = xi + vit + (1/2) a t2

v2 = vi2 + 2a (x - xi)

*where xi, vi refer to time = 0 s ;

x, v to time t

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Sample problem •  A brick is dropped (zero initial speed) from the roof of a building. The brick

strikes the ground in 2.50 s. Ignore air resistance, so the brick is in free fall. –  How tall, in meters, is the building? –  What is the magnitude of the brick’s velocity just before it reaches the ground? –  Sketch a(t), v(t), and y(t) for the motion of the brick.

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Sample problem •  A student standing on the ground throws a ball straight up. The ball leaves

the student’s hand with a speed of 15 m/s when the hand is 2.0 m above the ground. How long is the ball in the air before it hits the ground? –  Assume that the student moves her hand out of the way.

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Sample problem •  A cheetah spots a Thomson’s gazelle, its preferred prey, and leaps into

action, quickly accelerating to its top speed of 30 m/s, the highest of any land animal. However, a cheetah can maintain this extreme speed for only 15 s before having to let up. The cheetah is 170 m from the gazelle as it reaches top speed, and the gazelle sees the cheetah at just this instant. With negligible reaction time, the gazelle heads directly away from the cheetah, accelerating at 4.6 m/s2 for 5.0 s, then running at constant speed. Does the gazelle escape? If so, by what distance is the gazelle in front when the cheetah gives up?

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Sample problem •  A sprinter can accelerate with constant acceleration for 4.0 s before

reaching top speed. He can run the 100 meter dash in 10.0 s. What is his speed as he crosses the finish line?

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Motion in more than 1 dimension •  Have seen that 1D kinematics is written

in terms of quantities with a magnitude and a sign

•  Examples of 1D vectors

•  To extend to d > 1, we need a more general definition of vector

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Vectors: basic properties •  are used to denote quantities that have

magnitude and direction

•  can be added and subtracted

•  can be multiplied or divided by a number

•  can be manipulated graphically (i.e., by drawing them out) or algebraically (by considering components)

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Vectors: examples and properties •  Some vectors we will encounter:

position, velocity, force

•  Vectors commonly denoted by boldface letters, or sometimes arrow on top

•  Magnitude of A is written |A|, or no boldface and no absolute value signs

•  Some quantities which are not vectors: temperature, pressure, volume ….

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Drawing a vector •  A vector is represented graphically by a

line with an arrow on one end.

•  Length of line gives the magnitude of the vector.

•  Orientation of line and sense of arrow give the direction of the vector.

•  Location of vector in space does not matter -- two vectors with the same magnitude and direction are equivalent, independent of their location

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Adding vectors To add vector B to vector A

•  Draw vector A

•  Draw vector B with its tail starting from the tip of A

•  The sum vector A+B is the vector drawn from the tail of vector A to the tip of vector B.

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Multiplying vectors by a number •  Direction of vector not affected

(care with negative numbers – see below)

•  Magnitude (length) scaled, e.g. –  1*A=A

–  2*A is given by arrow of twice length, but same direction

–  0*A = 0 null vector

–  -A = -1*A is arrow of same length, but reversed in direction

A

1 x = A

2 x = A

0 x = A

-1 x = A

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Which of the vectors in the second row shows ?

2-2.2:

Slide 3-20

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Which of the vectors in the second row shows ?

QuickCheck 3.1

Slide 3-21

2-2.2:

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Which of the vectors in the second row shows 2 - ?

QuickCheck 3.2

Slide 3-27

2-2.3:

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Which of the vectors in the second row shows 2 - ?

QuickCheck 3.2

Slide 3-28

Clicker 2-2.3:

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Projection of a vector

“How much a vector acts along some arbitrary direction”

Component of a vector Projection onto one of the coordinate axes (x, y, z)

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Components

A

x

y

i

j

A = Ax + Ay

A = axi + ayj

i = unit vector in x direction

j = unit vector in y direction

ax, ay = components of vector A

Projection of A along coordinate axes

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More on vector components •  Relate components to

direction (2D):

ax = |A|cosθ,

ay = |A|sinθ or

•  Direction: tanθ = ay/ax

•  Magnitude: |A|2 = ax2 + ay

2

θ

A

x

y

Ay

Ax i

j

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2-2.4: A bird is flying along a straight line in a direction somewhere East of North. After the bird has flown a distance of 2.5 miles, it is 2 miles North of where it started. How far to the East is it from its starting point?

A.  0 miles B.  0.5 miles C. 1.0 mile D. 1.5 miles

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Why are components useful? •  Addition: just add components e.g. if C = A + B cx = ax + bx; cy = ay + by

•  Subtraction similar

•  Multiplying a vector by a number – just multiply components: if D = n*A dx = n*ax; dy = n*ay

•  Even more useful in 3 (or higher) dimensions

A x

y

i

j B

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Sample problem

Sam leaves his house and follows the following three step path: He heads 50.0 m due east. Then he travels 20.0m at 480 north of east. He then travels 70.0m 620 north of west.

Sketch the graph of Sam’s path in two-dimensions.

What is Sam’s net displacement?

What is the total distance that Sam travels?

If Sam moves at constant speed and the trip takes 100s, what is Sam’s speed? What is the magnitude of Sam’s average velocity?

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2D Motion in components

s – vector position → s = xQi + yQj

Note: component of position vector along x-direction is the x-coordinate!

x

y

Q

i

j

xQ

yQ

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Displacement in 2D Motion

s – vector position Displacement Δs = sF - sI, also a vector!

y

x O

sI

sF

Δs