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5.4 Integration & Anti-derivatives
Question #4 (Medium): Working With Acceleration, Velocity, Displacement, and Distance
Acceleration is the derivative of velocity. Velocity is the derivative of displacement.
Displacement is the net distance travelled, so
But Distance is the total distance travelled, so
The given acceleration (in and the initial velocity (in describe a particle moving along a line.
1) Find the velocity at time .
2) Find the total distance travelled during the given time interval.
1) Velocity is the anti-derivative of acceleration:
a. Since , then . So the constant factor
So the velocity function at time is: .
2) Note that the distance is not the net but total distance travelled:
Factoring , the -intercepts are
So over the intervals
and ; and over the interval
Split the interval according to these intervals:
Taking the anti-derivative, then plugging in endpoints of each subinterval:
Therefore, the total distance travelled during the time interval is .