Question

If a ball is thrown upward on the moon with a velocity of 18 m/s, its height s(t) in meters after t seconds is given by s(t) = 18t − 0.81t^2

 

(a) Write a formula for the average velocity of the ball for t near 2.

(b) Find the average velocity for the time interval beginning when t = 2 with duration 1

seconds, 0.5 seconds, and 0.05 seconds.

(c) What is your estimate for the instantaneous velocity of the ball at t = 2.

 

Answer 1

Step 1 :

(a)

The height of the ball after t seconds is .

Initial velocity of the ball at the surface of the Moon .

The average velocity of ball at time t is

The average velocity of ball at time t = 2 is

So the average velocity formula at t = 2 is .

Answer 2

Step 2 :

(b)

The average velocity for the time interval beginning when t = 2 with duration 1 seconds .

Substitute in average velocity formula .

The average velocity for the time interval beginning when t = 2 with duration 0.5 seconds .

Substitute in average velocity formula .

The average velocity for the time interval beginning when t = 2 with duration 0.05 seconds .

Substitute in average velocity formula .

Step 3 :

From the above calculations [ part (b) ] , one can estimate the instantaneous velocity at t = 2 .

When the duration of time decreases , the instantaneous velocity approaches to 14.7 m/s .

Check :

The instantaneous velocity at t = 2 is

Solution :

(a) the average velocity formula at t = 2 is .

(b) The average velocity when t = 2 with duration 1 seconds is .

     The average velocity when t = 2 with duration 0.5 seconds is .

     The average velocity when t = 2 with duration 0.5 seconds is .

(c) The instantaneous velocity at t = 2 is .