What is the average speed of an object that is moving at $4 m/s$ at $t=0$ and accelerates at a rate of $a(t) =2t^2-t$ on $t in [0,4]$ ?

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Answer 1 · 2017-11-06T06:54:16

The average speed is $=29/3ms^-1$

The speed is the integral of the acceleration

$a(t)=2t^2-t$

$v(t)=int(2t^2-t)dt=2/3t^3-1/2t^2+C$

The initial conditions are

$v=4ms^-1$ when $t=0$

Plugging these values in the equation of $v(t)$

$v(0)=2/3*0-1/2*0+C=4$

Therefore,

$C=4$

$v(t)=2/3t^3-1/2t^2+4$

The average speed is

$(4-0)barv=int_0^4(2/3t^3-1/2t^2+4)dt$

$4barv=[1/6t^4-1/6t^3+4t]_0^4$

$4barv=(128/3-8+4)-(0)$

$barv=1/4*116/3=29/3ms^-1$

What is the average speed of an object that is moving at 4 m/s4 m/s at t=0t=0 and accelerates at a rate of a(t) =2t^2-ta(t) =2t^2-t on t in [0,4]t in [0,4]?