Object moving position

Question

A moving object’s position, relative to some initial reference point, is given in metres by: s (t) = (t -4)^2 -6, where t is measured in seconds.

(a) Find the average velocity of the object over the time interval t = 0 to t = 10.

Using an appropriate limit, determine the instantaneous velocity of the object at t = 3.

Answer 1

To find the average velocity of the object over the time interval t = 0 to t = 10, we need to calculate the displacement and divide it by the time interval.

The displacement is given by the change in position, which can be found by evaluating s(t) at the endpoints of the interval:

s(10) = (10 - 4)^2 - 6 = 36 - 6 = 30

s(0) = (0 - 4)^2 - 6 = (-4)^2 - 6 = 16 - 6 = 10

The displacement is the difference between these two values:

Δs = s(10) - s(0) = 30 - 10 = 20

The time interval is t = 10 - 0 = 10.

Now we can calculate the average velocity:

Average velocity = Δs / Δt = 20 / 10 = 2 m/s

To find the instantaneous velocity at t = 3, we can take the derivative of the position function with respect to time:

s'(t) = 2(t - 4)

Now we can evaluate s'(t) at t = 3:

s'(3) = 2(3 - 4) = 2(-1) = -2 m/s

Therefore, the instantaneous velocity of the object at t = 3 is -2 m/s.

Please note that the unit of velocity is meters per second (m/s).