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A toy train starts from rest and is driven along a horizontal track by a motor which exerts a constant driving force. The effects of friction and air resistance can be neglected. Which of the following graphs best represents the momentum (M) of a train as a function of distance (S)?

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answer is C.

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Detailed Solution

The third graph depicts the relationship between the momentum of the particle and its displacement.
The external force acting on a body of mass m, according to Newton's second equation of motion, is given by:

F = \frac{dP}{dt}

Where

P = mv

is the momentum of the given body with velocity v. If the force and acceleration acting on a body are constant,

a = \frac{F}{m}, a constant

The velocity

v = 0 + at

And the displacement of the object is,

s = \frac{v^{2}}{2 a} or  s \propto v^{2}

.
Also the momentum of the object of mass m can be calculated as

P = mv or P \propto v

Hence the displacement:

s \propto v^{2} \propto P^{2}

Therefore the distance is proportional to the square of the momentum. The graph will be like:
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