Question

A motorcyclist who is moving along an x axis directed toward the east has an acceleration given by a = (6.30 - 2.20t) m/s2 for 0 ≤ t ≤ 6.0 s. At t = 0, the velocity and position of the cyclist are 3.00 m/s and 7.3 m. (a) What is the maximum speed achieved by the cyclist? (b) What total distance does the cyclist travel between t = 0 and 6.0 s?

Answer 1

(a) 12.02 m/s
 (b) 52.2 meters

   This problem is an example of integral calculus. You've been given an acceleration vector which is usually known as the 2nd derivative. From that you need to calculate the velocity function (1st derivative) and position (actual function) by successively calculating the anti-derivative. So:
   A(t) = 6.30 - 2.20t
 V(t) = 6.30t - 1.10t^2 + C

   We now have a velocity function, but need to determine C. Since we've been given the velocity at t = 0, that's fairly trivial.
 V(t) = 6.30t - 1.10t^2 + C
 3 = 6.30*0 - 1.10*0^2 + C
 3 = 0 + 0 + C
 3 = C

    So the entire velocity function is:
 V(t) = 6.30t - 1.10t^2 + 3
 V(t) = -1.10t^2 + 6.30t + 3

   Now for the location function which is the anti-derivative of the velocity function.
  V(t) = -1.10t^2 + 6.30t + 3
  L(t) = -0.366666667t^3 + 3.15t^2 + 3t + C

   Now we need to calculate C. And once again, we've been given the location for t = 0, so
 L(t) = -0.366666667t^3 + 3.15t^2 + 3t + C
 7.3 = -0.366666667*0^3 + 3.15*0^2 + 3*0 + C
 7.3 = 0 + 0 + 0 + C
 7.3 = C

   L(t) = -0.366666667t^3 + 3.15t^2 + 3t + 7.3

   Now that we have the functions, they are:
 A(t) = 6.30 - 2.20t
 V(t) = -1.10t^2 + 6.30t + 3
 L(t) = -0.366666667t^3 + 3.15t^2 + 3t + 7.3

    let's answer the questions.
 (a) What is the maximum speed achieved by the cyclist?
 This can only happen at those points that meet either of the following criteria.
  1. The derivative is undefined for the point.
  2. The value of the derivative is 0 for the point.
 As it turns out, the 1st derivative of the velocity function is the acceleration function which we have. So
 A(t) = 6.30 - 2.20t
 0 = 6.30 - 2.20t
  2.20t = 6.30
 t = 2.863636364

   So one of V(0), V(2.863636364), or V(6) will be the maximum value. Therefore: V(0) = 3
 V(2.863636364) = 12.0204545454545
 V(6) = 1.2

   So the maximum speed achieved is 12.02 m/s

   (b) Total distance traveled?
 L(0) = 7.3
 L(6) = 59.5
 Distance traveled = 59.5 m - 7.3 m = 52.2 meters