What is the average speed of an object that is still at #t=0# and accelerates at a rate of #a(t) = t^3-2t+2# from #t in [2, 3]#?

Question

What is the average speed of an object that is still at #t=0# and accelerates at a rate of #a(t) = t^3-2t+2# from #t in [2,   3]#?

Charlotte Aaron · Accepted Answer

#v_a=15 # #v_a=int_2^3( t^3-2*t+2)dt# #v_a=|3t^2-2|_2^3# #v_a=(3*3^2-2)-(3*2^2-2)# #v_a=(3*9-2)-(3*4-2)# #v_a=(27-2)-(12-2)# #v_a=25-(10)# #v_a=25-10# #v_a=15 #

Elena Albarran · Answer

undefined The average speed of an object that accelerates at a rate of $ a(t) = t^3 - 2t + 2 $ from $ t $ in [2, 3] can be calculated using the formula for average speed:

$$ \text{Average speed} = \frac{\text{Total distance}}{\text{Total time}} $$

To find the total distance traveled, integrate the velocity function over the given interval [2, 3]. Then divide the total distance by the total time, which is the difference between the final and initial times.

First, integrate the acceleration function $ a(t) = t^3 - 2t + 2 $ to find the velocity function $ v(t) $. Then integrate the velocity function over the interval [2, 3] to find the total distance traveled.

Next, calculate the total time, which is $ t_f - t_i = 3 - 2 = 1 $.

Finally, divide the total distance by the total time to find the average speed.

This process will provide the average speed of the object over the given time interval.