What are acceleration vs. time graphs?

The vertical axis represents the acceleration of the object.

For example, if you read the value of the graph shown below at a particular time, you will get the acceleration of the object in meters per second squared for that moment.

Try sliding the dot horizontally on the graph below to choose different times, and see how the acceleration—abbreviated Acc—changes.
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Concept check: According to the graph above, what is the acceleration at time $t=4\text{ s}$t, equals, 4, start text, space, s, end text?

The slope of an acceleration graph represents a quantity called the jerk. The jerk is the rate of change of the acceleration.

For an acceleration graph, the slope can be found from $\text{slope}=\dfrac{\text{rise}}{\text{run}}=\dfrac{a_2-a_1}{t_2-t_1}=\dfrac{\Delta a}{\Delta t}$start text, s, l, o, p, e, end text, equals, start fraction, start text, r, i, s, e, end text, divided by, start text, r, u, n, end text, end fraction, equals, start fraction, a, start subscript, 2, end subscript, minus, a, start subscript, 1, end subscript, divided by, t, start subscript, 2, end subscript, minus, t, start subscript, 1, end subscript, end fraction, equals, start fraction, delta, a, divided by, delta, t, end fraction, as can be seen in the diagram below.

This slope, which represents the rate of change of acceleration, is defined to be the jerk.

As strange as the name jerk sounds, it fits well with what we would call jerky motion. If you were in a ride where the acceleration was increasing and decreasing significantly over short periods of time, the motion would feel jerky, and you would have to keep applying different amounts of force from your muscles to stabilize your body.

To finish up this section, let's visualize the jerk with the example graph shown below. Try moving the dot horizontally to see what the slope—i.e., jerk—looks like at different points in time.

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Concept check: For the acceleration graph shown above, is the jerk positive, negative, or zero at $t=6\text{ s}$t, equals, 6, start text, space, s, end text?

The area under an acceleration graph represents the change in velocity. In other words, the area under the acceleration graph for a certain time interval is equal to the change in velocity during that time interval.

It might be easiest to see why this is the case by considering the example graph below which shows a constant acceleration of 4$~\dfrac{\text m}{\text s^2}$space, start fraction, start text, m, end text, divided by, start text, s, end text, squared, end fraction for a time of 9 s.

If we multiply both sides of the definition of acceleration, $a=\dfrac{\Delta v}{\Delta t}$a, equals, start fraction, delta, v, divided by, delta, t, end fraction, by the change in time, $\Delta t$delta, t, we get $\Delta v=a\Delta t$delta, v, equals, a, delta, t.

Plugging in the acceleration 4$~\dfrac{\text m}{\text s^2}$space, start fraction, start text, m, end text, divided by, start text, s, end text, squared, end fraction and the time interval 9 s we can find the change in velocity:

Multiplying the acceleration by the time interval is equivalent to finding the area under the curve. The area under the curve is a rectangle, as seen in the diagram below.

The area can be found by multiplying height times width. The height of this rectangle is 4$~\dfrac{\text m}{\text s^2}$space, start fraction, start text, m, end text, divided by, start text, s, end text, squared, end fraction, and the width is 9 s. So, finding the area also gives you the change in velocity.

The area under any acceleration graph for a certain time interval gives the change in velocity for that time interval.

A confident race car driver is cruising at a constant velocity of 20 m/s. As she nears the finish line, the race car driver starts to accelerate. The graph shown below gives the acceleration of the race car as it starts to speed up. Assume the race car had a velocity of 20 m/s at time $t=0\text{ s}$t, equals, 0, start text, space, s, end text.

What is the velocity of the race car after the 8 seconds of acceleration shown in the graph?

We can find the change in velocity by finding the area under the acceleration graph.

But this is just the change in velocity during the time interval. We need to find the final velocity. We can use the definition of the change in velocity, $\Delta v=v_f-v_i$delta, v, equals, v, start subscript, f, end subscript, minus, v, start subscript, i, end subscript, to find that

The final velocity of the race car was 44 m/s.

A sailboat is sailing in a straight line with a velocity of 10 m/s. Then at time $t=0\text{ s}$t, equals, 0, start text, space, s, end text, a stiff wind blows causing the sailboat to accelerate as seen in the diagram below.

The area under the graph will give the change in velocity. The area of the graph can be broken into a rectangle, a triangle, and a triangle, as seen in the diagram below.

The blue rectangle between $t=0\text{ s}$t, equals, 0, start text, space, s, end text and $t=3\text{ s}$t, equals, 3, start text, space, s, end text is considered positive area since it is above the horizontal axis. The green triangle between $t=3\text{ s}$t, equals, 3, start text, space, s, end text and $t=7\text{ s}$t, equals, 7, start text, space, s, end text is also considered positive area since it is above the horizontal axis. The red triangle between $t=7\text{ s}$t, equals, 7, start text, space, s, end text and $t=9\text{ s}$t, equals, 9, start text, space, s, end text, however, is considered negative area since it is below the horizontal axis.

We'll add these areas together—using $hw$h, w for the rectangle and $\dfrac{1}{2}bh$start fraction, 1, divided by, 2, end fraction, b, h for the triangles—to get the total area between $t=0\text{ s}$t, equals, 0, start text, space, s, end text and $t=9\text{ s}$t, equals, 9, start text, space, s, end text.

But this is the change in velocity, so to find the final velocity, we'll use the definition of change in velocity.

The final velocity of the sailboat is $v_f=28\text{ m/s}$v, start subscript, f, end subscript, equals, 28, start text, space, m, slash, s, end text.