Forces and Motion - Speed, Velocity, and Acceleration

A car accelerates from an initial velocity of $10\text{ m/s}$ to a final velocity of $25\text{ m/s}$ in a time interval of $5\text{ s}$. Calculate the acceleration.

$a = \frac{25\text{ m/s} - 10\text{ m/s}}{5\text{ s}} = 3\text{ m/s}^2$

Using the acceleration formula $a = \frac{v - u}{t}$, we subtract the initial velocity ($u$) from the final velocity ($v$) and divide by the time ($t$).

An athlete runs a $400\text{ m}$ race on a circular track and finishes at the exact same point they started in $50\text{ s}$. Calculate their average speed and average velocity.

$\text{Speed} = \frac{400\text{ m}}{50\text{ s}} = 8\text{ m/s}$, $\text{Velocity} = \frac{0\text{ m}}{50\text{ s}} = 0\text{ m/s}$

Average speed is total distance ($400\text{ m}$) divided by time. Velocity is displacement divided by time; since the athlete returned to the start, the displacement is $0\text{ m}$.

A train slows down from $30\text{ m/s}$ to a stop with a constant deceleration of $2\text{ m/s}^2$. How long does it take for the train to stop?

$t = \frac{v - u}{a} = \frac{0\text{ m/s} - 30\text{ m/s}}{-2\text{ m/s}^2} = 15\text{ s}$

We rearrange the acceleration formula to solve for time: $t = \frac{v - u}{a}$. Note that deceleration is expressed as a negative acceleration ($-2\text{ m/s}^2$).