class 051028

make sure you can do everything in Intro Set 4

Chapter 7 Problem Assignments:

121: 1, 2, 4, 11, 20, 31

1, 2, 4, 7, 11, 12, 15 - 20, 22, 25, 27, 32, 33

1.  A ball rolls off the edge of a table.  Its velocity is initially horizontal.  It falls 80 cm to the floor and has a horizontal range, from the edge of the table, of 70 cm.  How fast was it moving as it rolled off the table?

• Solve the vertical motion with `ds = -80 cm, a = -980 cm/s^2, v0 = 0.  You get around .4 seconds.
• Solve the horizontal motion with a = 0, `ds = 70 cm, `dt = .4 sec from the vertical motion.  Get v0 = vf = vAve = 170 cm/s.

The ball is now rolled off the edge of the table and collides with a second ball.  The collision is such that immediately after collision both balls are moving only in the horizontal direction.  Both balls fall 80 cm to the floor.  Their horizontal ranges are respectively 50 cm and 90 cm.  What were their velocities immediately after collision?

• Vertical motion is the same as in the preceding, since both initial velocities are in the horizontal direction.
• Horizontal velocities are 50 cm / .4 sec = 125 cm/s and 90 cm / .4 sec = 225 cm/s.

2.  A ball moving at 5 m/s collides with a ball moving at -2 m/s.  Immediately after collision both balls move in the same plane and along the same line as before, the first ball now moving at 1 m/s and the second at 6 m/s.

Which ball, if either, exerts the greater force on the other?

By Newton's Third Law the forces are equal and opposite.

Which ball is more massive?

The first ball changes velocity less than the second (more precisely the magnitude of the change in velocity is less for the first than for the second). 

Since forces are equal and opposite, the magnitude of the acceleration of the more massive object is less than the magnitude of the acceleration of the less massive.

So the magnitude of velocity change of the more massive object must be less than the magnitude of the change of the less massive.

Thus the first object must be the more massive.

Note that the velocity changes are

• `dv1 = 1 m/s - 5 m/s = -4 m/s, and
• `dv2 = 6 m/s - (-2 m/s) = 8 m/s.

By how much does the momentum of the first ball change during the collision?

Using m1 for the mass of the first and m2 for the mass of the second the momentum changes are

`dp1 = m1 * ( -4 m/s) and

`dp2 = m2 * 8 m/s.

By how much does the momentum of the second ball change during the collision?

What is the ratio of the masses of the balls?

Forces are equal and opposite so impulses are equal and opposite.

Impulse is equal to change in momentum.

So changes in momentum are equal and opposite.  Thus

`dp1 = - `dp2 or

m1 * (-4 m/s) = - m2 ( 8 m/s)

Thus

m2 / m1 = (-4 m/s) / (-8 m/s) = 1/2.

3.  A ball of mass .8 kg moving at 10 m/s collides with a ball of mass .2 kg moving at -6 m/s.  Immediately after collision both balls move in the same plane and along the same line as before, the first ball now moving at -4 m/s.

Which ball, if either, exerts the greater force on the other?

By Newton's Third Law the forces are equal and opposite.

By how much does the momentum of the first ball change?

The momentum of the first changes by

`dp1 = m1 ( v1 ' - v1 ) = .8 kg ( -4 m/s - 10 m/s) = -11.2 kg m/s.

By how much does the momentum of the second ball change?

The momentum change of the second ball is equal and opposite to that of the first so its momentum changes by +11.2 kg m/s.

By how much does the velocity of the second ball change?

The change in momentum of the second ball can be written in terms of its after-collision velocity as

`dp2 = m2 ( `dv2 ) = .2 kg ( `dv2 ) .

This must be equal to the +11.2 kg m/s change of the second ball.  So

.2 kg * `dv2 = 11.2 kg m/s and

`dv2 = 11.2 kg m/s / (.2 kg) = 56 m/s.

What is the final velocity of the second ball?

Since v2 = -6 m/s and `dv2 = 56 m/s, we get v2 ' = v2 + `dv2 = -6 m/s + 56 m/s = 50 m/s.

What is the definition of kinetic energy?

How do we find change in gravitational potential energy?

In a constant gravitational field with field strength g, the change in gravitational potential energy is m g `dy, where `dy is the change in vertical position.

What is the definition of work?

State the work-energy theorem.

What is the definition of impulse?

What is the definition of momentum?

How are impulse and momentum related?

When two objects collide how can you tell which exerts the greater force on the other?

Repeat the collision experiment we did last time but this time:

• Use carbon paper.
• Be sure you get everything you need to find the before- and after-collision velocities of both balls.
• Repeat each trial at least 5 times.

Now raise the ramp system a millimeter by placing metal straps under the ramps and repeat.  If the centers of the two balls were in a horizontal plane before, that will no longer be the case.  The goal is to determine if the difference due to that extra millimeter is detectable.