What is the average speed, on #t in [0,5]#, of an object that is moving at #12 m/s# at #t=0# and accelerates at a rate of #a(t) =5-3t# on #t in [0,4]#?
The acceleration of an object is simply the derivative of its velocity with respect to time, so the velocity of an object is the integral of the acceleration with respect to time, in this case
Where we have added the arbitrary constant
To get the average velocity, the simplest thing to do is to find the distance traveled in the specified time, which is the integral of the velocity between the limits:
The average velocity is then the total distance over the total time: