(a) A player lobs a mid-court pass with a 624-g basketball, which covers 15 m in 2 s. What is the basketball’s horizontal translational kinetic energy while in flight? (b) An average molecule of air, in the basketball in part (a), has a mass of 29 u, and an average speed of 500 m/s, relative to the basketball. There are about
molecules inside it, moving in random directions, when the ball is properly inflated. What is the average translational kinetic energy of the random motion of all the molecules inside, relative to the basketball? (c) How fast would the basketball have to travel relative to the court, as in part (a), so as to have a kinetic energy equal to the amount in part (b)?
Strategy
In part (a), first find the horizontal speed of the basketball and then use the definition of kinetic energy in terms of mass and speed,
. Then in part (b), convert unified units to kilograms and then use
to get the average translational kinetic energy of one molecule, relative to the basketball. Then multiply by the number of molecules to get the total result. Finally, in part (c), we can substitute the amount of kinetic energy in part (b), and the mass of the basketball in part (a), into the definition
, and solve for
v .
Solution
The horizontal speed is (15 m)/(2 s), so the horizontal kinetic energy of the basketball is
The average translational kinetic energy of a molecule is
and the total kinetic energy of all the molecules is
Significance
In part (a), this kind of kinetic energy can be called the horizontal kinetic energy of an object (the basketball), relative to its surroundings (the court). If the basketball were spinning, all parts of it would have not just the average speed, but it would also have rotational kinetic energy. Part (b) reminds us that this kind of kinetic energy can be called internal or thermal kinetic energy. Notice that this energy is about a hundred times the energy in part (a). How to make use of thermal energy will be the subject of the chapters on thermodynamics. In part (c), since the energy in part (b) is about 100 times that in part (a), the speed should be about 10 times as big, which it is (76 compared to 7.5 m/s).
Summary
The kinetic energy of a particle is the product of one-half its mass and the square of its speed, for non-relativistic speeds.
The kinetic energy of a system is the sum of the kinetic energies of all the particles in the system.
Kinetic energy is relative to a frame of reference, is always positive, and is sometimes given special names for different types of motion.
Conceptual questions
A particle of
m has a velocity of
Is its kinetic energy given by
If not, what is the correct expression?
One particle has mass
m and a second particle has mass 2
m . The second particle is moving with speed
v and the first with speed 2
v . How do their kinetic energies compare?
The first particle has a kinetic energy of
whereas the second particle has a kinetic energy of
so the first particle has twice the kinetic energy of the second particle.
A person drops a pebble of mass
from a height
h , and it hits the floor with kinetic energy
K . The person drops another pebble of mass
from a height of 2
h , and it hits the floor with the same kinetic energy
K . How do the masses of the pebbles compare?
(a) How fast must a 3000-kg elephant move to have the same kinetic energy as a 65.0-kg sprinter running at 10.0 m/s? (b) Discuss how the larger energies needed for the movement of larger animals would relate to metabolic rates.
Estimate the kinetic energy of a 90,000-ton aircraft carrier moving at a speed of at 30 knots. You will need to look up the definition of a nautical mile to use in converting the unit for speed, where 1 knot equals 1 nautical mile per hour.
Calculate the kinetic energies of (a) a 2000.0-kg automobile moving at 100.0 km/h; (b) an 80.-kg runner sprinting at 10. m/s; and (c) a
electron moving at
