Question

A ball is thrown into the air by a baby alien on a planet in the system of Alpha Centauri with a velocity of 46 ft/s. Its height in feet after t seconds is given by y=46t−19t2. A. Find the average velocity for the time period beginning when t=3 and lasting

.01 s:
.005 s:
.002 s:
.001 s:

Answer 1

The average velocities are:

for 0.01.s

• 
  `<v> = - 68.19 (ft)/(s) \hat{j}`

for 0.005 s

• 
  `<v> = - 68.2 (ft)/(s) \hat{j}`

for 0.002 s

• 
  `<v> = - 68.05 (ft)/(s) \hat{j}`

for 0.001 s

• 
  `<v> = - 68.02 (ft)/(s) \hat{j}`

Step-by-step explanation:

The average velocity is given by

`<v> = (displacement)/(time)`

So, we just need to find the position at t = 3 s and then, after every period of time.

Position at t = 3 s

Knowing that

`y(t) = 46 (ft)/(s) t - 19 (ft)/(s^2) \ t^2`

at t = 3 s we have

`y(3 \ s) = 46 (ft)/(s) * 3 s - 19 (ft)/(s^2) * (3 \ s)^2`

`y(3 \ s) = 138 ft - 19(ft)/(s^2) * 9 \ s^2`

`y(3 \ s) = -33 ft`

After 0.01 s

After 0.01 s the position will be

`y(3.01 s) = 46 (ft)/(s) 3.01 s - 19 (ft)/(s^2) \ (3.01 s)^2`

`y(3.01 s) = -33.6819 ft`

So, the average velocity will be

`<v> = \frac{-33.6819 ft \hat{j}- (-33 ft) \hat{j}}{0.01 s}`

`<v> = \frac{-0.6819 ft \hat{j}}{0.01 s}`

`<v> = - 68.19 (ft)/(s) \hat{j}`

The minus sign is there cause the velocity is pointing downward.

After 0.005 s

After 0.005 s the position will be

`y(3.005 s) = 46 (ft)/(s) 3.005 s - 19 (ft)/(s^2) \ (3.005 s)^2`

`y(3.005 s) = -33.3405 ft`

So, the average velocity will be

`<v> = \frac{-33.3405 ft \hat{j} - (-33 ft) \hat{j} }{0.005 s}`

`<v> = (-0.3405 ft )/(0.005 s) \hat{j}`

`<v> = - 68.2 (ft)/(s) \hat{j}`

After 0.002 s

After 0.002 s the position will be

`y(3.002 s) = 46 (ft)/(s) 3.002 s - 19 (ft)/(s^2) \ (3.002 s)^2`

`y(3.002 s) = -33.1361 ft`

So, the average velocity will be

`<v> = \frac{-33.1361  ft \hat{j} - (-33 ft) \hat{j} }{0.002 s}`

`<v> = \frac{-0.1361  ft \hat{j} }{0.002 s}`

`<v> = - 68.05 (ft)/(s) \hat{j}`

After 0.001 s

After 0.001 s the position will be

`y(3.001 s) = 46 (ft)/(s) 3.001 s - 19 (ft)/(s^2) \ (3.001 s)^2`

`y(3.001 s) = -33.06802 ft`

So, the average velocity will be

`<v> = \frac{-33.06802  ft  \hat{j} - (-33) ft \hat{j} }{0.001 s}`

`<v> = \frac{-0.06802  ft \hat{j} }{0.001 s}`

`<v> = - 68.02 (ft)/(s) \hat{j}`