Acceleration Graph

#217 / Dynamics Easy

The below graph plots the acceleration of a motorcycle racing on a track. If the bike started from rest, what is its speed at the 7 seconds mark?

Expand Hint

Speed is the velocity integral with respect to time:

$$$Speed=initial\:speed+\int_{0}^{t}a\:dt$$$

Hint 2

By solving an integral, you are calculating the area under the curve. Thus, we can break down the graph by segments:

Speed is the velocity integral with respect to time:

$$$Speed=initial\:speed+\int_{0}^{t}a\:dt$$$

$$$=0+\int_{0}^{2}a\:dt+\int_{2}^{4}a\:dt+\int_{4}^{6}a\:dt-\int_{6}^{7}a\:dt$$$

By solving an integral, you are calculating the area under the curve. Thus, we can break down the graph by segments:

0-2 is a triangle: $$\frac{1}{2}bh=\frac{1}{2}(2)(2)=2$$
2-4 is a rectangle: $$bh=2(2)=4$$
4-6 is a triangle: $$\frac{1}{2}bh=\frac{1}{2}(2)(2)=2$$
6-7 is a triangle (in the negative direction): $$-\frac{1}{2}bh=-\frac{1}{2}(1)(1)=-\frac{1}{2}$$

Combining all the segments together to find the speed at 7 seconds from initial launch:

$$$0+2+4+2-\frac{1}{2}=7.5\:m/s$$$

7.5 m/s

Time Analysis

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