Set 2 Problem number 1

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• Problem

• Solution

• Generalized Solution

• Explanation in terms of Figure(s), Extension

• Figure(s)

Problem

The velocity of an object increases at a constant rate of 6 meters/second per minute. How long will it take the object to increase its velocity by 18 meters/second?

Solution

The velocity would go up by 6 meters per second in 1 minute, by 12 meters per second two minutes, etc..

If we continue counting in units of 6 meter/sec, each corresponding to a minute of acceleration, we see that velocity increases by 18 m/s in 3 min.

We might also say that the rate of velocity increase is 6 m/s per minute, or 6 (m/s) / min. 

• Velocity is the changing quantity. 
• The change in velocity is thus the product of the rate and the time interval over which the rate acts, or

• change in velocity = rate of change of velocity * time interval
• the required time interval is thus equal to   change in velocity / rate of change of velocity
• In symbols we would say that    `dv = 6 (m/s) / min * `dt
• It follows that `dt = `dv / [ 6 (m/s) / min] = 18 m/s / [ 6 (m/s) / min]  = 3 minutes.

Generalized Solution

Generalized Response: If we let a stand for the number of meters per second by which the velocity increases in a minute, and `dt the number of minutes, then the velocity increase in `dt minutes will be the product a `dt of the velocity increase per minute and the number of minutes.

If we represent velocity increase by `dv, we thus have

    `dv = a `dt.

We understand a as the rate at which velocity increases. We call a the acceleration of the object.

Explanation in terms of Figure(s), Extension

Figure description:

The figure below uses arrows to represent four different velocities, each greater than the one before it, and each corresponding to a time 1 minute later than the one before it. Longer arrows represent greater velocities.

You can see that the velocities increase by the same amount during each minute. The short red arrow represents the velocity change during the first minute. The long red arrow represents the total velocity change during the three minutes represented by the picture. Since the velocity change is the same for every minute, the total velocity change is 3 times as great as the per-minute change.

If a represents the velocity change per minute, then the total 3-minute velocity change `dv is

`dv = 3 * a.

Figure(s)