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

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

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Answer 1 · 2016-02-08T05:01:16

12.5

Explanation:

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

$v (t) = v (0) + \int_{0}^{t} a (τ) d τ$ $v(t) = v(0) + int_0^t a(tau) d tau$

$= 8 + \int_{0}^{t} (5 - 2 τ) d τ$ $= 8 + int_0^t (5-2tau) d tau$

$= 8 + {[5 τ - τ^{2}]}_{0}^{2}$ $= 8 + [5tau-tau^2]_0^t$

$= 8 + 5 t - t^{2}$ $= 8 + 5t - t^2$

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

$x (t) - x (0) = \int_{0}^{t} v (τ) d τ$ $x(t) - x(0) = int_0^t v(tau) d tau$

$= \int_{0}^{t} (8 + 5 τ - τ^{2}) d τ$ $= int_0^t (8 + 5tau - tau^2) d tau$

$= {[8 τ + \frac{5}{2} τ^{2} - \frac{1}{3} τ^{3}]}_{0}^{2}$ $= [8tau + 5/2tau^2 - 1/3tau^3]_0^t$

$= 8 t + \frac{5}{2} t^{2} - \frac{1}{3} t^{3}$ $= 8t + 5/2t^2 - 1/3t^3$

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

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

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

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

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

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Explanation:

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What is the average speed of an object that is moving at 8ms8 m/s at t=0t=0 and accelerates at a rate of a(t)=5−2ta(t) =5-2t on t∈[0,3]t in [0,3]?

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Explanation:

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