From a bridge high above a river, ball A is thrown straight up with initial speed vi. Ball B is thrown straight down with the same initial speed, vi.Each hits the water. Compare their impact speeds.

1. vimpact.A > vimpact.B

2. vimpact.A < vimpact.B

3. vimpact.A = vimpact.B

4. There is not enough information (“it depends”).

10/7/15 1Oregon State University PH 211, Class #5

A ball is dropped from rest from the same height from which a bullet is fired horizontally over level ground. Neglecting any effects of air or topography, which would hits the ground first?

1. The bullet hits the ground first.

2. The ball hits the ground first.

3. They both hit at the same time.

10/7/15 2Oregon State University PH 211, Class #5

The motion in each dimension acts independent of the other.

10/7/15 3Oregon State University PH 211, Class #5

Sketch what you see—for each of these observations:

• The motion of the cart with respect to the table

• The motion of the ball with respect to the cart

• The motion of the ball with respect to the table

10/7/15 4Oregon State University PH 211, Class #5

10/7/15 Oregon State University PH 211, Class #5 5

A car passes by you at a constant velocity of 20 m/s (over level ground), just as you fire a pellet from a gun. The pellet leaves the gun at an initial speed of 40 m/s. At what initial angle (above the horizontal) would the gun need to be aimed in order for the pellet to hit the car?

1. 30°2. 60 °3. 15 °4. 45 °5. None of the above.

10/7/15 Oregon State University PH 211, Class #5 6

2-Dimensional Motion (with Constant Acceleration)

The four equations of kinematics help us describe and calculate the motion of any object undergoing constant acceleration (including zero acceleration) with respect to any single axis.

But what if that object is moving relative to two axes at once?The vector nature of displacement, velocity and acceleration let us calculate the x- and y- motions separately.

For the motion along each axis, we use the respective vector components (x, vx.i, vx.f, and ax or y, vy.i, vy.f, and ay).

The one consistent connection between the two parts of the motion is time: We’re talking about one object, so the same time interval, t, applies to both x- and y- motions.

10/7/15 Oregon State University PH 211, Class #5 7

x-motion y-motion

vx.f = vx.i+ax(t) vy.f = vy.i+ay(t)

x = (1/2)(vx.i+vx.f)t y = (1/2)(vy.i+vy.f)t

x = vx.i(t)+(1/2)ax(t)2 y = vy.i(t)+(1/2)ay(t)2

vx.f2 = vx.i

2+2ax(x) vy.f2 = vy.i

2+2ay(y)

(t is common to both motions.)

ax is constant ay is constant

10/7/15 Oregon State University PH 211, Class #5 8

In all cases, the actual motion of the object is the vector sum of the component motions. For example, starting with ti = 0, xi = 0 and yi = 0, notice how completely you can describe the motion:

The object’s actual position at any time, t, is a vector sum: x + y(Examples: What is the pellet’s position at t = 0 s? 3 s? 4s? 8s?)

The object’s actual velocity at time t is a vector sum: vx + vy

(Examples: What is the pellet’s velocity at t = 0 s? 3 s? 4s? 8s?)

The object’s actual acceleration at time t is a vector sum: ax + ay

(Examples: What is the pellet’s acceleration at t = 0 s? 3 s? 4s? 8s?)

You kick a soccer ball at a 45° angle.It goes up and down and hits the ground. What can you say about its speed at thetop of its arc?

1. It is 9.8 m/s2

2. It is 0 m/s

3. It is 9.8 m/s

4. It is faster than immediately after the kick.

5. It is slower than immediately after the kick.

10/7/15 9Oregon State University PH 211, Class #5

A baseball player friend of yours wants to determine how fast she can throw a baseball. You have her stand on a flat roof and throw the ball horizontally. The ball is released 4m above the ground and it lands 25 meters away. How fast did she throw the ball? How fast did it hit the ground?

10/7/15 10Oregon State University PH 211, Class #5