Question

A train car with a mass of 1 kg and speed of 11 m/s is traveling to the right. Another train car with a mass of 2 kg is moving at -2 m/s (to the left). After the collision, the 1 kg train car is now moving at -1 m/s and we need to find the Velocity of the 2 kg train car.

Answer 1

Before the collision, the total linear momentum of the system before the collision is:

`P=p_1+p_2`

Where p₁ and p₂ are the linear momenta of the cars with masses 1kg and 2kg respectively:

`\begin{gathered} p_1=m_1v_1 \\ p_2=m_2v_2 \end{gathered}`

After the collision, the cars exchange linear momenta and the total linear momentum of the system becomes:

`P^(\prime)=p_1^(\prime)+p_2^(\prime)`

According to the Law of Conservation of Linear Momentum, the total linear momentum of the system remains constant before and after the collision. Then:

`\begin{gathered} P=P^(\prime) \\ \\ \Rightarrow m_1v_1+m_2v_2=m_1v_1^(\prime)+m_2v_2^(\prime) \end{gathered}`

The initial velocities of both cars are given, as well as the masses of the cars and the final velocity of the 1kg train car. Then, the only unknown variable in the above equation is v₂', the final velocity of the 2kg train car. Isolate v₂' from the equation:

`\begin{gathered} m_1v_1+m_2v_2=m_1v_1^(\prime)+m_2v_2^(\prime) \\ \\ \Rightarrow m_2v_2^(\prime)=m_1v_1+m_2v_2-m_1v_1^(\prime) \\ \\ \Rightarrow v_2^(\prime)=(m_1v_1+m_2v_2-m_1v_1^(\prime))/(m_2) \end{gathered}`

Consider that the velocities directed to the left are negative and those directed to the right are positive. Then, replace m₁=1kg, m₂=2kg, v₁=11m/s, v₁'=-1m/s and v₂=-2m/s to find the final velocity of the 2kg train car:

`\begin{gathered} v_2^(\prime)=(m_1v_1+m_2v_2-m_1v_1^(\prime))/(m_2) \\ \\ \Rightarrow v_2^(\prime)=((1kg)(11(m)/(s))+(2kg)(-2(m)/(s))-(1kg)(-1(m)/(s)))/(2kg) \\ \\ \Rightarrow v_2^(\prime)=(11kg(m)/(s)-4kg(m)/(s)+1kg(m)/(s))/(2kg) \\ \\ \Rightarrow v_2^(\prime)=(8kg(m)/(s))/(2kg) \\ \\ \Rightarrow v_2^(\prime)=4(m)/(s) \end{gathered}`

Therefore, the final velocity of the 2kg train car is 4 meters per second to the right.