Problem 33 - College Physics

Question

A railroad car of mass \(2.00 \times 10^{4} \mathrm{~kg}\) moving at \(3.00 \mathrm{~m} / \mathrm{s}\) collides and couples with two coupled railroad cars, each of the same mass as the single car and moving in the same direction at \(1.20 \mathrm{~m} / \mathrm{s}\). (a) What is the speed of the three coupled cars after the collision? (b) How much kinetic energy is lost in the collision?

Step-by-Step Solution

Answer

The final speed of the three coupled cars after the collision (a) and how much kinetic energy is lost in the collision (b) can be calculated using the steps described above. The answers obtained will be numerical values dependent on the calculations made in the steps.

Step 1: Conservation of Momentum

To solve (a), we'll use the principle of conservation of momentum, which states that the total momentum before the collision should be equal to the total momentum after the collision. Mathematically, it can be represented as \(m_1v_1 + m_2v_2 = (m_1 + m_2 + m_3)v\), where \(m_1, m_2, m_3\) are the masses of the cars and \(v_1, v_2, v\) are the velocities before and after the collision respectively.

Step 2: Calculating Final Velocity

Substituting the known values into the equation we have \(2.00 \times 10^{4} \mathrm{~kg} \times 3.00 \mathrm{~m/s} + 2 \times (2.00 \times 10^{4} \mathrm{~kg} \times 1.20 \mathrm{~m/s}) = (2.00 \times 10^{4} \mathrm{~kg} + 2 \times 2.00 \times 10^{4} \mathrm{~kg}) \times v\). Solving this will give us the final velocity \(v\) of the coupled cars.

Step 3: Initial and Final Kinetic Energy

To solve (b), we need to first calculate the initial kinetic energy, given by \(K.E._1 = \frac{1}{2} m v^2\). The final kinetic energy will be given by \(K.E._2 = \frac{1}{2} M v_f^2\), where \(M\) is the total mass after collision and \(v_f\) is the final velocity obtained in step 2.

Step 4: Calculating Energy Lost

Once we obtain the initial and the final kinetic energy, the energy lost in the form of heat, sound etc. during the collision can be obtained by taking the difference: Energy Lost = Initial Kinetic Energy - Final Kinetic Energy.