## The First Problem

*A car has to travel a distance on a straight road. The car has maximum acceleration and maximum deceleration . It starts and ends at rest.*

*Show that if there is no speed limit, the time is given by*

.

*Solution: *We are going to use two pieces of information:

- the area under the velocity-time graph is the distance travelled,
- the slope of the velocity-time graph is the acceleration.

First, a rough sketch of the velocity-time graph:

*The slope of the first line segment is , while the slope of the second line segment is . The area under the curve is given by , the total distance. The base is , the total time, and the height is the maximum speed.*

Let be the maximum speed reached. Using the area of a triangle, where is the total time:

.

Now using the fact that the slope of the line segments, rise/run, is equal to the accelerations, we derive:

, and

.

Thus

.

Now, using :

.

Taking square roots (and noting ) completes the proof.

## The Second Problem (1997)

*A particle, moving in a straight line, accelerates uniformly from rest to a speed . It continues at this constant speed for a time and then decelerates uniformly to rest, the magnitude of the deceleration being twice that of the acceleration. The distance traveled while accelerating is 6 m. The total distance traveled is 30 m and the total time taken is 6 s. *

i. *Draw a velocity-time graph and hence, or otherwise, find .*

ii. *Calculate the distance travelled at constant velocity.*

*Solution: *

i. First the velocity-time graph:

There are number of things we need to use:

- the area under the velocity-time graph is the distance traveled
- the slope of the velocity-time graph is the acceleration

The question gives the following:

- the total distance is 30 m
- the total time is 6 s
- the distance traveled from to is 6 m.

In the first triangle:

.

In the second triangle:

,

and also . Define Let so that . Therefore, using the fact that the total time is 6 s:

.

Now, using the fact that the area under the first triangle is 6:

,

and the total area is 30:

m/s.

ii. Using we have so that the time spent at constant velocity is:

,

and therefore the distance traveled is:

.

Alternatively note that and so m.

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