Question

A motor scooter travels east at a speed of \(12 \mathrm{m} / \mathrm{s}\). The driver then reverses direction and heads west at \(15 \mathrm{m} / \mathrm{s}\) What is the change in velocity of the scooter? Give magnitude and direction.

Short answer

Answer: The change in velocity is \(27 \mathrm{m/s}\) towards the west.

Step-by-step solution

Represent the given velocities with their directions

We are given the initial velocity of the motor scooter as \(12 \mathrm{m/s}\) towards the east and the final velocity as \(15 \mathrm{m/s}\) towards the west. Let's represent east as positive and west as negative. So the initial velocity, \(v_i\), is \(+12 \mathrm{m/s}\), and the final velocity, \(v_f\), is \(-15 \mathrm{m/s}\).

Calculate the change in velocity

Now we find the change in velocity, denoted as \(\Delta v\). The change in velocity is the difference between the final velocity and the initial velocity: \(\Delta v = v_f - v_i\) Substitute the values of \(v_i\) and \(v_f\): \(\Delta v = (-15 \mathrm{m/s}) - (+12 \mathrm{m/s})\)

Calculate the result

Now we can perform the subtraction: \(\Delta v = -15 \mathrm{m/s} - 12 \mathrm{m/s} = -27 \mathrm{m/s}\)

Interpret the result

The change in velocity is \(-27 \mathrm{m/s}\). The negative sign indicates that the change in velocity is in the west direction. In terms of magnitude and direction, the change in velocity is \(27 \mathrm{m/s}\) towards the west.