Question

Two parallel rail tracks run North-South. Train $A$ moves due North with a speed of $54 km h^{- 1}$ and train $B$ moves due South with a speed of $90 km h^{- 1}$ . A monkey runs on the roof of the train $A$ with a velocity of $18 km h^{- 1}$ with respect to the train $A$ , in a direction opposite to that of $A$ . Calculate
$(a)$ the relative velocity of $B$ with respect to $A$ .
$(b)$ the relative velocity of the ground with respect to $B$ .
$(c)$ the velocity of a monkey as observed by a man, standing on the ground.
$(d)$ the velocity of the monkey as observed by a passenger of the train $B$ .

Accepted Answer

$54 km h^{- 1}$ First, we want to calculate the velocity of $B$ , with respect to $A$ , $v_{BA} = v_{B} - v_{A}$ .

Now, by taking the sign convention, train $A$ is moving in the North direction, so the sign will be positive.

The train $18 km h^{- 1}$ is moving in the South direction, so the sign will be negative,

$\Rightarrow v_{B} = - 90 km h^{- 1}$ and $v_{A} = 54 km h^{- 1}$

$(d)$

$= - 144 km h^{- 1}$ .

The answer is $144 km h^{- 1}$ in the South direction.

$(b)$ The relative velocity of the ground with respect to $B$ ,

$v_{GB} = v_{G} - v_{B} = 0 - 90$

$v_{GB} = - 90 km h^{- 1}$ .

It means the relative velocity of the ground with respect to $B$ is $90 km h^{- 1}$ , in the direction of North because the train is moving in the South direction.

$(c)$ Now, the relative velocity of the ground with respect to the monkey is $v_{MG}$ ,
$(c)$ .

The relative velocity of ground with respect to $A$ , and relative velocity of the monkey with respect to $A$ ,

$v_{M} - v_{G} = v_{M} + v_{A} - v_{A} - v_{G} = v_{M} - v_{A} + v_{A} - v_{G}$

$\Rightarrow v_{MG} = v_{MA} + v_{AG}$ .

The monkey is moving in the negative $x$ -direction with respect to $A$ , in the North direction,

$\Rightarrow v_{MG} = - 18 + 54$

$\Rightarrow v_{MG} = 36 km h^{- 1}$ .

$(d)$ The passenger of the train has the velocity of the train $B$ , so the relative velocity of the monkey with respect to the passenger,

$v_{MB} = v_{M} - v_{B} = - 36 - 90 = - 126 km h^{- 1}$ .

It means the relative velocity of the monkey with respect to the passenger is $126 km h^{- 1}$ , in the North direction.

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