Physics 581 Midterm

Monday, August 3, 1998 11 - 1

1. (a) Give one sentence descriptions of Galileo’s "natural horizontal motion" and "natural vertical motion" and show with a diagram how he combined these ideas to describe projectile motion.

(b) Taking g = 10 meters per sec per sec, find the average speed of a falling object during the first half-second of fall, and figure out from that how far it fell during that time.

(c) An earthquake in California leaves a chasm 12 meters wide running at right angles to a road. The pavement is level on each side of the chasm, but one side is 1.25 meters lower than the other side. How fast must a car coming from the high side have to be going on approaching the chasm to make it across?

2. A child is bouncing up and down on a trampoline. When no-one is on the trampoline, it is level. When the child stands on the trampoline, it sags 0.3 meters. Assume in this question that the child is simply bouncing—that is, neglect the energy the child is feeding in to maintain the motion.

In drawing the graphs below, just make very rough estimates of how far you expect the trampoline to sag at maximum depth, and how high the child bounces.

(a) Draw a fairly careful graph of how the child’s *acceleration *varies through one bounce cycle.

(b) Draw another graph of the child’s *position* as a function of time through one cycle. This graph must be drawn *directly below* the first graph, with the *same* time scale, so that the position at a given instant is directly below the acceleration at that instant.

3. The moon takes 27.3 days to travel once around the earth, in an orbit close to a circle of radius 384,000 km.

(a) From these data, find its speed in km per second.

(b) Find how far the moon’s orbit falls away below a straight line in one second. Do this by drawing a triangle, one vertex being at the center of the earth, another being where the moon is at some instant, the third where it would be one second later if it were moving in a straight line. Deduce the value of the earth’s gravitational field at the moon’s orbit in terms of the field strength here at the earth’s surface.

4. A child of mass 20 kg is swinging on a swing having rope of length 2 meters. The maximum angle of swing is 30 degrees from the vertical.

(a) What is the maximum potential energy of the child, relative to being at rest?

(b) What is the maximum speed of the child?

(c) What is the maximum tension in the rope?

(d) Show on a diagram the direction and approximate magnitude of the child’s acceleration at the midpoint of the swing, and at the far point. Show the direction of the acceleration at some other point, mentioning also which way the swing is moving at that point.

5. A pool ball traveling across the table at speed *v* hits another ball having the same mass, which was initially at rest. The first ball does not come to rest.

(a) Prove that the velocities of the balls after the collision are perpendicular to each other.

(b) Does it follow from Newton’s Third Law that the change in velocity of one of the balls must be the negative of the change in velocity of the other? Show with a vector diagram what these changes are.

(c) Show with a diagram how the center of mass of the two balls moves before the collision and after the collision.

(d) Describe how the collision would appear to someone moving with the center of mass.