Set 1 Problem number 4

• Problem

• Solution

• Generalized Solution

• Explanation in terms of Figure(s), Extension

• Figure(s)

Problem

How long does it take an object moving at 8 meters/second to move 40 meters?

Solution

Moving at 8 meters every second, we would go 8 meters in 1 second, 16 meters in 2 seconds, 24 meters in 3 seconds, . . ., 40 meters in 40 seconds. It helps to visualize this.

• After visualizing it becomes clear that we can just divide the 40 meters by 8 meters / second to get the number of seconds required.
• We also note that when we divide meters by   meters/second we get seconds (m / (m/s) = m * (s / m) = (m * s) / m = s).

Generalized Solution

Generalized Response: Since displacement, or position change `ds is equal to the product of average velocity and time interval,

`ds = vAve `dt,

the time interval must be

`dt = `ds / vAve.

This is a simple algebraic rearrangement of `ds = vAve `dt (we just divide both sides by vAve), and it is also common sense: if we moved 6 meters at 2 meters per second we would require 3 seconds.

• This is obvious if we think about it with obvious numbers (e.g., 20 meters at 5 m/s), and we observe that when the numbers are obvious we naturally divide the displacement by the time interval.
• We do the same thing when the numbers aren't obvious.

Explanation in terms of Figure(s), Extension

Figure description:

The image below shows a 'triangle' connecting vAve, `dt and `ds. This triangle indicates that vAve,`dt and `ds are connected in such a way that the value of any of the three quantities can be found from the values of the other two.

We know that `ds = vAve `dt. If we divide both sides of this equation by `dt and reverse sides we get

vAve = `ds / `dt.

If we divide both sides by vAve and reverse sides we get

`dt = `ds / vAve.

These relationships are indicated on the diagram.

This specific problem could be solved by direct reasoning. We can also get the answer using the relationship `dt = `ds / vAve, with the given `ds and vAve.

Figure(s)