Question

An object moves along the x-axis according to the

equation x(t)= 3t2 - 2t + 3 meters. Determine
A. The average velocity between t= 2 s and t= 3 s. (b)
B. The instantaneous velocity at t= 3 s.
C. The instantaneous acceleration at t= 3 s.

Answer 1

The average velocity between t=2 s and t=3 s is 15 m/s, the instantaneous velocity at t=3 s is 16 m/s, and the instantaneous acceleration at t=3 s is 6 m/s².

Step-by-step explanation:

The equation x(t) = 3t2 - 2t + 3 meters describes the motion of an object along the x-axis.

(a) To determine the average velocity between t = 2 s and t = 3 s, we find the position of the object at these times using the position equation and then divide the change in position by the change in time.

Average velocity = ∆x / ∆t = [x(3) - x(2)] / (3 - 2) = [(3(3)2 - 2(3) + 3) - (3(2)2 - 2(2) + 3)] / (3 - 2) = (24 - 9) / 1 = 15 m/s

(b) The instantaneous velocity at any time t can be found by taking the derivative of the position function. v(t) = d[x(t)]/dt = d(3t2 - 2t + 3)/dt = 6t - 2. Thus, the instantaneous velocity at t = 3 s is v(3) = 6(3) - 2 = 16 m/s.

(c) The instantaneous acceleration is the derivative of the velocity function. a(t) = d[v(t)]/dt = d(6t - 2)/dt = 6. Therefore, the instantaneous acceleration at t = 3 s is 6 m/s2.