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Chapter 9: Problem 29

In a totally inelastic collision between two equal masses, with one initially at rest, show that half the initial kinetic energy is lost.

Short Answer

Expert verified
Upon solving, we observe that the final kinetic energy of the system is half of the initial kinetic energy. Therefore, in a totally inelastic collision between two equal masses (with one initially at rest), half the initial kinetic energy is indeed lost.

Step by step solution

01

Calculate Initial Kinetic Energy of the System

Before the collision, one object is moving while the other is at rest. Let's denote the moving object's velocity as \(v_i\) and both objects' mass as \(m\). Therefore, the initial kinetic energy \(K_i\) of the system is only the kinetic energy of the moving object, given by \( K_i = \frac{1}{2} m (v_i)^2 \).
02

Use Conservation of Momentum

According to the conservation of momentum, the total momentum before the collision equals the total momentum after the collision. Thus, we can find the final velocity \(v_f\) of the combined mass after the collision using the equation \( m v_i = 2m v_f \). From this, we derive \(v_f = \frac{v_i}{2}\).
03

Compute Final Kinetic Energy of the System

After the collision, the two objects stick together and move as one object of mass \(2m\). To find the final kinetic energy \(K_f\), we use the expression \( K_f = \frac{1}{2} (2m) v_f^2 = \frac{1}{2} (2m) (\frac{v_i}{2})^2 \). Simplifying, we find \( K_f = \frac{1}{2} m (v_i)^2 \). Hence, the final kinetic energy is half of the initial kinetic energy.

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