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Acceleration

The acceleration of an object is its change in velocity divided by the time the change takes.

acceleration = change in velocity\time interval:

The graph below might be the change in velocity of a skier who is moving downhill at a constant acceleration:

Graph of velocity/time

The skier starts at time = 0 with a velocity = 0. After 1 second, the velocity is 0.5 m/s. After 2 seconds, the velocity has increased to 1 m/s. For every second, the velocity increases by 0.5 m/s. We can say that the change in velocity is constant and equal to 0.5 m/s every second. This can be written as 0.5 m/s/s, or 0.5 m/s2.

Displacement = Area under the Velocity-Time curve

The area under the velocity-time graph is the displacement. The above graph of constant acceleration forms a triangle between the velocity curve and the time axis. The area of a triangle is half its height times its base: 0.5 x 6 m/s x 12 s = 36m.

Acceleration = Slope of the Velocity-Time curve

The slope of the velocity-time graph is the acceleration. The steeper the slope, the greater the acceleration. To find the slope, choose any two points: slope =

Case Study: traffic lights

Consider the motion of a car between traffic lights:

Animation of car at traffic lights

What would this motion look like in a graph of velocity against time?

Graph of velocity/time

Question: What is the acceleration in the first two seconds?

Answer: initial velocity .
Final velocity .
Therefore,
Using the formula

Note that 30 km/h can be expressed as m/s, by dividing by 3.6

Question: What is the acceleration in the last two seconds?

Answer: initial velocity .
Final velocity .
Therefore,
Using the formula
The acceleration is equal in magnitude to phase A, but opposite in direction (deceleration).

Question: What is the total displacement travelled by the car in the ten seconds?

Answer: the displacement is the area under the velocity curve. This is in three parts, A, B, and C.
First we need to convert 60 km/h into m/s so that the unit of time matches.
60 km/h /3.6 = 16.7 m/s
Area A: 0.5 ⋅ v ⋅ t = 0.5 ⋅ 16.7 m/s ⋅ 2 s = 16.7 m
Area C is the same as A
Area B: in this case, there is zero acceleration, but constant velocity. In this case, we use the formula
Total displacement = total area =

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