Question

A particle moves along the x-axis according to x(t)=10t−2t2m. (a) What is the instantaneous velocity at t = 2 s and t = 3 s? (b) What is the instantaneous speed at these times? (c) What is the average velocity between t = 2 s and t = 3 s?

a) 1. (a) 30 m/s, 10 m/s; 2 m/s, 5 m/s; 5 m/s, 10 m/s; 10 m/s, 30 m/s
b) 2. (b) 10 m/s, 5 m/s; 10 m/s, 5 m/s; 5 m/s, 10 m/s; 30 m/s, 10 m/s
c) 3. (c) 15 m/s, 7.5 m/s; 7.5 m/s, 3.75 m/s; 3.75 m/s, 7.5 m/s; 7.5 m/s, 15 m/s

Answer 1

The instantaneous velocity at t = 2 s is 2 m/s and at t = 3 s is -2 m/s. The instantaneous speed at these times is 2 m/s. The average velocity between t = 2 s and t = 3 s is 22 m/s.

Step-by-step explanation:

(a) To find the instantaneous velocity at t = 2 s and t = 3 s, we can take the derivative of the position function x(t). The derivative of 10t - 2t^2 is 10 - 4t. So, at t = 2 s, the velocity is 10 - 4(2) = 2 m/s, and at t = 3 s, the velocity is 10 - 4(3) = -2 m/s.

(b) Instantaneous speed is the magnitude of the instantaneous velocity. So, at t = 2 s, the speed is |2| = 2 m/s, and at t = 3 s, the speed is |-2| = 2 m/s.

(c) Average velocity is the displacement divided by the time interval. The displacement between t = 2 s and t = 3 s can be found by evaluating x(3) - x(2) = [(10(3) - 2(3)^2) - (10(2) - 2(2)^2)] = 22 m. The time interval is 3 s - 2 s = 1 s. So, the average velocity is 22 m/1 s = 22 m/s.