Chapter 9, Problem 52A

A hammer falls off a rooftop and strikes the ground with a certain KE. If it fell from a roof that was four times higher, how would its KE of impact compare? Its speed of impact? (Neglect air resistance.)

To determine

The comparison of the kinetic energy of impact and the speed of impact with the initial values when the height of the rooftop increases by four times.

Answer to Problem 52A

The kinetic energy will become four times of the initial kinetic energy while the velocity will be two times of the initial value.

Explanation of Solution

Introduction:

As per law of conservation of energy, the energy of an isolated system cannot be destroyed or created. It always converts from one form to the another.

The gravitational potential energy is given by,

  $PE=mgh$

Where,

  $m=$ mass

  $g=$ gravity

  $h=$ height

And, the kinetic energy is given by,

  $KE=\frac{1}{2}m{v}^2$

Where,

  $v=$ velocity

Since the hammer was at rest at the rooftop, hence its kinetic energy will be zero there. Similarly, the potential energy of the hammer will be zero at the floor. Now, considering the law of conservation of energy,

  $\begin{array}{l}PE=KE\\ \Rightarrow KE\propto h\end{array}$

Hence, if the height of the rooftop becomes four times, then the kinetic energy will also become four times of its initial value.

Now, it is known from the above mentioned formula,

  $v\propto \sqrt{KE}$

Therefore, if the kinetic energy becomes four times of its initial value, the velocity becomes twice of the its initial value.

Conclusion:

Hence, the kinetic energy will be four times while the velocity will be twice of there initial values.