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Chapter 15: Problem 38

A railroad car having a mass of \(15 \mathrm{Mg}\) is coasting at \(1.5 \mathrm{~m} / \mathrm{s}\) on a horizontal track. At the same time another car having a mass of \(12 \mathrm{Mg}\) is coasting at \(0.75 \mathrm{~m} / \mathrm{s}\) in the opposite direction. If the cars meet and couple together, determine the speed of both cars just after the coupling. Find the difference between the total kinetic energy before and after coupling has occurred, and explain qualitatively what happened to this energy.

Short Answer

Expert verified
After calculating all variables as explained, remember that the result might depend on the specific values for the speed and mass. The kinetic energy difference accounts for energy lost to other forms such as thermal or sound energy and potential deformation of the cars during the collision.

Step by step solution

01

Calculate Initial Momentum

The initial momentum of the system is calculated as the sum of the momentum of both cars before the collision. Momentum is calculated with the formula \(p=mv\) where \(m\) is the mass and \(v\) is the velocity. Hence, the initial momentum, \(p_{i}\), is \(p_{i}=(15 Mg \cdot 1.5 m/s) + (12 Mg \cdot -0.75 m/s)\). The negative sign is applied because the second car is moving in the opposition direction.
02

Calculate Final Velocity

Under the conservation of momentum, the total momentum before and after the collision should be equal. So after the collision where the cars couple together, we treat them as one object with a mass of \(15 Mg + 12 Mg = 27 Mg\). Representing the final velocity of this object as \(v\), we equate the initial and final momentum to get \(p_{i} = (27 Mg) \cdot v\) and solve for \(v\).
03

Calculate Kinetic Energy Before Collision

Kinetic energy is found using the formula \(KE=\frac{1}{2}mv^2\). Therefore, the total kinetic energy before the collision, \(KE_{i}\), can be found by adding the kinetic energy of the two cars: \(KE_{i}=\frac{1}{2}(15 Mg \cdot (1.5 m/s)^2) + \frac{1}{2}(12 Mg \cdot (0.75 m/s)^2)\).
04

Calculate Kinetic Energy After Collision

After the collision, both cars move together with the same velocity which was found in Step 2. We calculate the total kinetic energy after the collision, \(KE_{f}\), using the same formula as before but with the combined mass and shared velocity: \(KE_{f}=\frac{1}{2}(27 Mg \cdot v^2)\).
05

Find the Difference in Kinetic Energy

The difference in kinetic energy is simply the absolute value of the difference between the initial and final kinetic energy: \(|KE_{i} - KE_{f}|\). This difference represents energy that was transferred to other forms during the coupling. A qualitative explanation of where this energy goes might involve discussion of heat and sound energy from the collision, deformations in the cars at the point of contact, amongst others.

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Most popular questions from this chapter

A ballistic pendulum consists of a \(4-\mathrm{kg}\) wooden block originally at rest, \(\theta=0^{\circ}\). When a 2 -g bullet strikes and becomes embedded in it, it is observed that the block swings upward to a maximum angle of \(\theta=6^{\circ} .\) Estimate the initial speed of the bullet.

The cart has a mass of \(3 \mathrm{~kg}\) and rolls freely from \(A\) down the slope. When it reaches the bottom, a spring loaded gun fires a \(0.5-\mathrm{kg}\) ball out the back with a horizontal velocity of \(v_{b / c}=0.6 \mathrm{~m} / \mathrm{s}\), measured relative to the cart. Determine the final velocity of the cart.

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