Q. 17 Three identical train cars, coup... [FREE SOLUTION]

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Chapter 11: Q. 17 (page 287)

Three identical train cars, coupled together, are rolling east at speed $v_{o}$ . A fourth car traveling east at $2 v_{o}$ catches up with the three and couples to make a four-car train. A moment later, the train cars hit a fifth car that was at rest on the tracks, and it couples to make a five-car train. What is the speed of the five-car train?

Short Answer

Expert verified

The firth car's celocity is $v_{o}$ .

Step by step solution

Given information

We need to find the speed of the five-car train?

Simplify

When the initial momentum equals the final momentum, the momentum of an isolated system changes to zero. There is no interaction with the environment within the system.The momentum is conserved, and we therefore use equation (11.23) in the form of conservation of momentum

${\overset{鈫�}{p}}_{f} = {\overset{鈫�}{p}}_{i} (1)$

A mass and speed object has momentum when they move with speed, and momentum is a vector, which is the product of mass and velocity.

$P = m v (2)$

First case:

To move with velocity $v_{o}$ that coupled with three identical cars and same mass car moves with velocity $2 v_{o}$ the initial momentum of the system is

$p_{i} = (3 m) (v_{o}) + (m) (2 v_{o}) = 5 m v_{o}$

The four cars are coupled and moved with velocity $v_{f}$ , $1$ after collision and momentum after collision is

$P_{f} = 4 m v_{f, 1}$

Using equation $(1)$ for expresion of $v_{f}$ , $1$ in the form

${\overset{鈫�}{p}}_{f} = {\overset{鈫�}{p}}_{f} 4 m v_{f, 1} = 5 m v_{o} v_{f, 1} = \frac{5}{4} v_{o} {v_{f,}}_{1} = 1.25 v_{o}$

Simplify

The four cars coupled and moves with velocity $1.25 v_{o}$ and the fifth cars is at rest $v_{i}, 2 = 0$ .Before collision the initial momentum of the system is

$p_{i} = (4 m) (1.25 v_{o}) + (m) (0) = 5 m v_{o}$