Question 238: Work, Energy and Power & Circular Motion

Question

A body is dropped on ground from a height '$$h_{1}$$' and after hitting the ground, it rebounds to a height '$$h_{2}$$'. If the ratio of velocities of the body just before and after hitting ground is 4 , then percentage loss in kinetic energy of the body is $$\frac{x}{4}$$. The value of $$x$$ is ____________.

Accepted Answer

The velocity of the body just before hitting the ground, due to gravitational acceleration, is given by $$v_{1} = \sqrt{2gh_{1}}$$, and the velocity just after hitting the ground, when it rebounds to a height $$h_{2}$$, is given by $$v_{2} = \sqrt{2gh_{2}}$$.
According to the problem, the ratio $$\frac{v_{1}}{v
{2}} = 4$$. Therefore, we can write $$\frac{\sqrt{2gh_{1}}}{\sqrt{2gh_{2}}} = 4$$ or equivalently $$\frac{h_{1}}{h_{2}} = 4^2 = 16$$.
The loss in kinetic energy due to the collision with the ground is given by the difference between the initial kinetic energy $$K_{1} = \frac{1}{2} m v_{1}^2$$ and the final kinetic energy $$K_{2} = \frac{1}{2} m v_{2}^2$$, where m is the mass of the body.
Substituting $$v_{1} = \sqrt{2gh_{1}}$$ and $$v_{2} = \sqrt{2gh_{2}}$$ into these expressions, we get $$K_{1} = mgh_{1}$$ and $$K_{2} = mgh_{2}$$.
The loss in kinetic energy is then $$\Delta K = K_{1} - K_{2} = mgh_{1} - mgh_{2}$$.
The percentage loss in kinetic energy is given by
$$\frac{\Delta K}{K_{1}} 	imes 100 = \frac{mgh_{1} - mgh_{2}}{mgh_{1}} 	imes 100 = \frac{h_{1} - h_{2}}{h_{1}} 	imes 100$$.
Since $$h_{1}/h_{2} = 16$$, we can write $$h_{2} = h_{1}/16$$, so the percentage loss in kinetic energy is
$$\frac{h_{1} - h_{1}/16}{h_{1}} 	imes 100 = 100(1 - \frac{1}{16}) = 100 	imes \frac{15}{16} = \frac{375}{4}$$.
So, the value of $$x$$ is 375.

Work, Energy and Power & Circular Motion

A body is dropped on ground from a height '$h_{1}' and after hitting the ground, it rebounds to a height 'h_{2}'. If the ratio of velocities of the body just before and after hitting ground is 4 , then percentage loss in kinetic energy of the body is \frac{x}{4}. The value of x$ is ____________.

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