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Question: An empty freight car of mass\({m_1} = 10,000kg\)rolls on a level road and collides with a loaded car...

An empty freight car of massm1=10,000kg{m_1} = 10,000kgrolls on a level road and collides with a loaded car of mass standing at rest. If the cars couple together, find their speed after the collision and also the loss in kinetic energy. What should be the speed of the loaded car towards the empty car, in order that both be brought to rest by the collision?

Explanation

Solution

In order to calculate the speed of the loaded car towards the empty car, we need to use the formula of momentum conservation, m1u1=(m1+m2)v{m_1}{u_1} = ({m_1} + {m_2})v
According to this the momentum of a body remains constant; it is neither created nor destroyed.

Step by step solution,
By momentum conservation,
m1u1=(m1+m2)v{m_1}{u_1} = ({m_1} + {m_2})v
v=23ms−1v = \dfrac{2}{3}m{s^{ - 1}}
Kinetic Energy loss=12m1u12−12(m1+m2)v2 = \dfrac{1}{2}{m_1}u_1^2 - \dfrac{1}{2}({m_1} + {m_2}){v^2}
12(10000×22−(30000)(23)2)\dfrac{1}{2}\left( {10000 \times {2^2}\quad - \left( {30000} \right){{\left( {\dfrac{2}{3}} \right)}^2}} \right)
=12×10000(4−43)= \dfrac{1}{2} \times 10000\left( {4 - \dfrac{4}{3}} \right)
=12×83×10000=\dfrac{1}{2}\times \dfrac{8}{3}\times 10000
=13333.34J= 13333.34J
Thus, the correct answer to this question is 13333.34J13333.34J

Additional Information: Collision means two objects coming into contact with each other for a very short period. In other words, collision is a reciprocate interaction between two masses for a very short interval wherein the momentum and energy of the colliding masses changes. While playing carroms, you might have noticed the effect of a striker on coins when they both collide. Collision is any event in which two or more bodies exert forces on each other in about a relatively short time. Although the most common use of the word collision refers to incidents in which two or more objects collide with great force, the scientific use of the term implies nothing about the magnitude of the force.

Note: While solving this question, one should be familiar with the concept of momentum conservation and the formulam1u1=(m1+m2)v{m_1}{u_1} = ({m_1} + {m_2})v. Some concepts are provided above for the reference.
We have to keep in mind that u stands for initial speed and v stands for final speed.