Question Detail
Question
The motion of a particle from O, is described by the equations S = (2/3)t3 - (17/2)t2 + 21t where S is the distance in metres, and t the time in seconds. Find the acceleration of the particle when it is momentarily at rest.
Correct Answer
Option B is the correct answer.
Detailed Explanation
- Find the velocity (v) by differentiating S with respect to t: S = (2/3)t3 - (17/2)t2 + 21t v = dS/dt = 2t2 - 17t + 212. Find when the particle is momentarily at rest (v = 0): 2t2 - 17t + 21 = 0 This is a quadratic equation. We can solve it by factoring or using the quadratic formula. Using factoring: (2t - 3)(t - 7) = 0 So, t = 3/2 seconds or t = 7 seconds.3. Find the acceleration (a) by differentiating v with respect to t: a = dv/dt = d/dt (2t2 - 17t + 21) = 4t - 174. Evaluate acceleration at the times when the particle is at rest: At t = 3/2: a(3/2) = 4(3/2) - 17 = 6 - 17 = -11 ms-2 At t = 7: a(7) = 4(7) - 17 = 28 - 17 = 11 ms-2Since the options include 11 ms-2, this is one of the possible accelerations when the particle is at rest.
Hint
First, find the velocity function. Set velocity to zero to find the times when the particle is at rest. Then, find the acceleration function and evaluate it at these times.