Question

If particle A at ${\mathrm{t}}_1=-2.0\mathrm{s}$ is at ${\mathrm{x}}_1=4.8\mathrm{cm}$, and at ${\mathrm{t}}_2=4.5\mathrm{s}$ is at ${\mathrm{x}}_2=8.5\mathrm{cm}$, what is its average velocity over this time interval? Can you calculate its average speed from these data? Why or why not?

Short answer

The average velocity over the time interval is $0.57\frac{\mathrm{cm}}{\mathrm{s}}$.

No, you can’t calculate the average speed from these data.

Here, you have only the initial and final positions for the corresponding time, and you don’t know the path the particle has traveled in, from the initial to the final position. So, you can only calculate the displacement and are unable to calculate the distance during this time interval. You can’t find the average speed from these given data.

Step-by-step solution

Step 1. Relation between the velocity, positions, and time

Given data.

The initial position is $x_1=4.8\mathrm{cm}$.

The final position is $x_2=8.5\mathrm{cm}$.

The initial time is $t_1=-2.0\mathrm{s}$.

The final time is $t_2=4.5\mathrm{s}$.

You know that $\mathrm{Average}\mathrm{velocity}=\frac{\mathrm{final}\mathrm{position}-\mathrm{initial}\mathrm{position}}{\mathrm{time}\mathrm{elapsed}}$.

Step 2. Calculation of the average velocity with the given values

Now, substituting the values,

$\begin{array}{c}\mathrm{Average}\mathrm{velocity}=\frac{x_2-x_1}{t_2-t_1}\\ =\frac{8.5\mathrm{cm}-4.8\mathrm{cm}}{4.5\mathrm{s}-\left(-2.0\mathrm{s}\right)}\\ =0.57\frac{\mathrm{cm}}{\mathrm{s}}\end{array}$.

Therefore, the average velocity over the time interval is $0.57\frac{\mathrm{cm}}{\mathrm{s}}$.

Step 3. Reason why speed cannot be found from this data

No, we can’t calculate the average speed from these data.

Here, you have only the initial and final positions for the corresponding time, and you don’t know which path the particle has traveled in, from the initial to the final position.

So, you can only calculate the displacement for this time interval and not the distance for the same. Therefore, finding the average speed from this data is not possible.