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

Answer 1

Charlotte Aaron

The average speed is #=3.53ms^-1#

The speed is the integral of the acceleration

#v(t)=inta(t)dt#

#=int(2t^2-2)dt#

#=2/3t^3-2t+C#

Plugging in the initial conditions

#v(0)=0-0+C=3#

Therefore,

#v(t)=2/3t^3-2t+3#

#v(2)=2/3*2^3-2*2+3=13/3#

The average value is

#(5-0)barv=int_0^2(2/3t^3-2t+3)dt+13/3*3#

#5barv=[2/3*1/4*t^4-2/2t^2+3t]_0^2+13#

#=(8/3-4+6)-(0)+13#

#=14/3+13=53/3#

#barv=53/15=3.53ms^-1#

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Answer 2

Adrian Ayala

To find the average speed, integrate the velocity function over the given time interval and divide by the total time. The average speed is ( \frac{1}{5} \left[6t - \frac{2}{3}t^3 \right]_0^5 ). After evaluating, the average speed is 21 m/s.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.