Question

if all else is constant, which would cause the greatest increase in kinetic energy of a moving object?
mass and velocity are each reduced by one-half.
mass is reduced by one-half, and velocity is doubled.
mass is doubled, and velocity is reduced by one-half.
mass and velocity are each doubled.

Explanation:

The formula for kinetic energy is \( KE=\frac{1}{2}mv^{2} \), where \( m \) is mass and \( v \) is velocity.

• For option A: New \( KE_A=\frac{1}{2}(\frac{m}{2})(\frac{v}{2})^{2}=\frac{1}{2}\times\frac{m}{2}\times\frac{v^{2}}{4}=\frac{1}{16}mv^{2} \) (decrease).
• For option B: New \( KE_B=\frac{1}{2}(\frac{m}{2})(2v)^{2}=\frac{1}{2}\times\frac{m}{2}\times4v^{2}=mv^{2} \).
• For option C: New \( KE_C=\frac{1}{2}(2m)(\frac{v}{2})^{2}=\frac{1}{2}\times2m\times\frac{v^{2}}{4}=\frac{1}{4}mv^{2} \) (decrease).
• For option D: New \( KE_D=\frac{1}{2}(2m)(2v)^{2}=\frac{1}{2}\times2m\times4v^{2}=4mv^{2} \).

Comparing \( KE_B = mv^{2} \) and \( KE_D = 4mv^{2} \), doubling both mass and velocity gives the greatest increase in kinetic energy.

Answer:

D. Mass and velocity are each doubled.