What is the average speed of an object that is moving at $8 m/s$ at $t=0$ and accelerates at a rate of $a(t) =5-2t$ on $t in [0,3]$ ?

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Answer 1 · 2016-02-08T05:01:16

12.5

Integrate the acceleration to get the velocity as a function of time.

$v(t) = v(0) + int_0^t a(tau) d tau$

$= 8 + int_0^t (5-2tau) d tau$

$= 8 + [5tau-tau^2]_0^t$

$= 8 + 5t - t^2$

Integrate the velocity to get the displacement as a function of time.

$x(t) - x(0) = int_0^t v(tau) d tau$

$= int_0^t (8 + 5tau - tau^2) d tau$

$= [8tau + 5/2tau^2 - 1/3tau^3]_0^t$

$= 8t + 5/2t^2 - 1/3t^3$

From this, we can calculate the displacement from $t=0$ to $t=3$ .

$x(3)-x(0) = 8(3) + 5/2(3)^2 - 1/3(3)^3 = 37.5$

Divide this by the time (3 seconds) to get the average velocity.

$bar(v) = frac{37.5}{3-0} = 12.5$

What is the average speed of an object that is moving at 8 m/s8 m/s at t=0t=0 and accelerates at a rate of a(t) =5-2ta(t) =5-2t on t in [0,3]t in [0,3]?