Question

A train is traveling south at 31.2 m/s when the brakes are applied. It slows down with constant acceleration to a speed of 6.00 m/s in a time of 9.00 s.What is the acceleration of the train during the 9.00-s interval? If the acceleration is toward north, enter a positive value. If the acceleration is toward south, enter a negative value.How far does the train travel during the 9.00 s? If the displacement is toward north, enter a positive value. If the displacement is toward south, enter a negative value.

Answer 1

Acceleration:
`2.80\; {\rm m\cdot s^(-2)}` (towards the north.)

Displacement: approximately
`(-167)\; {\rm m}` (towards the south.)

Step-by-step explanation:

The initial velocity
`u` of the train (before braking) will be negative since the the train was travelling south. Thus:
`u = (-31.2)\; {\rm m\cdot s^(-1)}`.

The train continues southwards after braking. Therefore, the final velocity
`v` of the train (after braking) will also be negative:
`v = (-6.00)\; {\rm m \cdot s^(-1)}`.

Find the change in the velocity of the train by subtracting the initial value from the final value:

`\begin{aligned}\Delta v &= v - u \\ &= (-6.00)\; {\rm m\cdot s^(-1) - (-31.2)\; {\rm m\cdot s^(-1) \\ &= 25.2\; {\rm m\cdot s^(-1)} \end{aligned}`.

Divide change in velocity by the time required
`t = 9.00\; {\rm s}` to find the (constant) acceleration
`a` of the train:

`\begin{aligned} a &= (\Delta v)/(t) \\ &= \frac{25.2\; {\rm m\cdot s^(-1)}}{9.00\; {\rm s}} \\ &= 2.80\; {\rm m\cdot s^(-2)}\end{aligned}`.

Note that acceleration is positive since the velocity is becoming less negative. The velocity component of the train towards the south has been reduced.

Apply the SUVAT equation
`(v^(2) - u^(2)) = 2\, a\, x` to find the displacement
`x` of the train:

`\begin{aligned} x &= (v^(2) - u^(2))/(2\, a) \\ &= ((-6.00)^(2) - (-31.2)^(2))/(2 * 2.80)\; {\rm m} \\ &\approx (-167)\; {\rm m} \end{aligned}`.

Note that displacement is negative since the train keeps travelling south. The position of the train after braking is to the south of where the train was before braking.