A particle moves on the x-axis so that its velocity at any time t is given by v(t) = sin 2t. At t = 0, the particle is at the origin.

Question

a)For 0 ≤ t ≤ π, find all values of t for which the particle is moving to the left.
b)Write an expression for the position of the particle at any time t.
c)For 0 ≤ t ≤ π/2, find the average value of the position function determined in part b.

Answer 1

a)For 0 ≤ t ≤ π, the particle is moving to the left when sin 2t < 0. This occurs when t = π/2, 3π/2, 5π/2, etc.

b)The position of the particle at any time t is given by the integral of the velocity function, which is x(t) = -1/2 cos 2t.

c)For 0 ≤ t ≤ π/2, the average value of the position function is given by the integral of x(t) from 0 to π/2, divided by π/2. This is equal to -1/4.