Question

a moving objects kinetic energy depends on its mass. if a baseball and a tennis ball move at the same speed, the ball with more mass has more kinetic energy.
look at the figure below. note the vertical bars. these are energy bars. energy bars show relative amounts of energy. the more full the bar, the more energy the object has. the tennis ball and the baseball are traveling at the same speed, but the baseball has a greater mass. for objects traveling at the same speed, the more mass an object has, the greater its kinetic energy.
the relationship between mass and kinetic energy is a proportional, linear relationship that can be described mathematically.

three-dimensional thinking
for each system determine how the quantity of kinetic energy would increase or decrease.

1. if the mass of an object is increased by a factor of 4, kinetic energy increases by a factor of ______________.
2. if the mass of an object is decreased by a factor of $\frac{1}{2}$, kinetic energy decreases by a factor of ______________.

Explanation:

Step1: Understand the relationship between KE and mass

The formula for kinetic energy is \(KE=\frac{1}{2}mv^{2}\). When speed \(v\) is constant, \(KE\propto m\) (kinetic energy is directly proportional to mass).

Step2: Solve for the first case

If \(m_{new} = 4m_{old}\), and \(KE=\frac{1}{2}mv^{2}\) (with \(v\) constant), then \(KE_{new}=\frac{1}{2}(4m_{old})v^{2}\). Since \(KE_{old}=\frac{1}{2}m_{old}v^{2}\), \(KE_{new} = 4KE_{old}\).

Step3: Solve for the second case

If \(m_{new}=\frac{1}{2}m_{old}\), and \(KE=\frac{1}{2}mv^{2}\) (with \(v\) constant), then \(KE_{new}=\frac{1}{2}(\frac{1}{2}m_{old})v^{2}\). Since \(KE_{old}=\frac{1}{2}m_{old}v^{2}\), \(KE_{new}=\frac{1}{2}KE_{old}\).

Answer:

1. \(4\)
2. \(\frac{1}{2}\)