Problem 5.12

Suppose you toss a tennis ball upward.

(a) Does the kinetic energy of the ball increase or decrease as it moves higher?

Kinetic energy (KE) is energy that an object possesses by virtue of it's mass and the fact that it is in motion. KE is given by

KE = 1/2mv2

Where m is the object's mass (in kg) and v is velocity ( in m/sec.) Recall that our SI unit of energy is the Joule, which is another term for kg m2/sec2. See where that comes from now? Anyway, the total energy of an object can always be written as the sum of the KE and the object's potential energy (PE) (see below) as

In other words, the total energy E of an object in motion subject to some sort of interaction with another object is a constant. What does this mean?? Well, if we throw a tennis ball upward, the tennis ball has both KE and PE, and their sum is always a constant. The PE of the ball must increase (see part b), so if PE increases, then KE must decrease (because the sum is a constant.) The ball slows down as it reaches it's highest point (at which KE =0 and all energy is potential), and falls back to earth. As the ball falls back, the PE goes down and the KE goes up, so the ball moves faster until it's back on the ground. (b) What happens to the potential energy of the ball as it moves higher?

Potential energy (PE) is energy that an object possesses by virtue of it's position relative to another object with which it can interact. This interaction can be due to gravitational forces (which depend on mass and distance), electrostatic forces (which depend on charges and distance), or mechanical forces (think about two masses connected by a spring.) In this case, we're dealing with the interaction between the tennis ball and the earth. When the distance between the ball and the earth is zero (i.e., the ball is sitting on the ground), the PE is zero. As the distance between the two objects increases, the PE increases, so as the ball moves upwards, PE increases.

(c) If the same amount of energy were imparted to a ball the same size as a tennis ball, but of twice the mass, how high would it go in comparison to the tennis ball?

Think about this one a little: We impart the same amount of total energy E to both balls, and E = KE + PE. Here, we need to know the form of the PE - it is given by mgh, where m is mass, g is the gravitational constant, and h is height. At the very top of the trajectory of the ball, all KE has been converted into PE (this is the point at which the ball stops and reverses its direction.) OK, since E is the same for both balls, and since the mass of the second ball is twice the mass of the first, then the heavier ball must only go half as high as the first.

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Doug Chapman 7/8/08