Question

Displacement from velocity The following functions describe the velocity of a car (in mi/hr) moving along a straight highway for a 3-hr interval. In each case, find the function that gives the displacement of the car over the interval [0,t], where 0 ≤ t ≤ 3.
v(t) = { 30 if 0 ≤ t ≤ 2
50 if 2 < t < 2.5
44 if 2.5 < t ≤ 3
Graph showing velocity in mi/hr over time in hours, with distinct segments indicating changes in speed.

Graph showing velocity in mi/hr over time in hours, with distinct segments indicating changes in speed.

Accepted Answer

Understand that displacement is the integral of velocity over time. Since velocity is given as a piecewise constant function, the displacement function will be a piecewise linear function obtained by integrating each velocity segment over its respective time interval. For the interval $$0 \leq t \leq 2$$, the velocity is constant at 30 mi/hr. The displacement function here is the integral of 30 with respect to $$t$$, which is $$30t$$ plus a constant of integration. Since displacement at $$t=0 is $$zero, the constant is zero, so $$s(t) = 30t$$ for $$0 \leq t \leq 2. $$For the interval $$2 \u003c t \leq 2.5$$, the velocity changes to 50 mi/hr. The displacement function here is the displacement at $$t=2$$ plus the integral of 50 from 2 to $$t. $$Calculate the displacement at $$t=2$$ using the previous piece: $$s(2) = 30 	imes 2 = 60. $$Then, for $$2 \u003c t \leq 2.5$$, $$s(t) = 60 + 50(t - 2). $$For the interval $$2.5 \u003c t \leq 3$$, the velocity is 44 mi/hr. Similarly, find the displacement at $$t=2.5$$ using the previous piece: $$s(2.5) = 60 + 50(2.5 - 2) = 60 + 50 	imes 0.5 = 85. $$Then, for $$2.5 \u003c t \leq 3$$, $$s(t) = 85 + 44(t - 2.5). $$Combine these piecewise expressions to write the full displacement function $$s(t)$$ over the interval $$0 \leq t \leq 3. $$This function gives the total displacement of the car from time 0 up to any time $$t in $$the interval.

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Key Concepts

Displacement as the Integral of Velocity

Piecewise Functions and Integration

Interpreting Graphs of Piecewise Velocity Functions