This is confusing me

Question

Please explain HOW so I know what to do

If a ball is thrown straight up with an initial velocity of 70ft/sec. Its height after t seconds is given by y=70t-16t^2. Find the average velocity for the time period beginning when t=1 and lasting

0.1 sec

0.01 sec

0.001 sec

Based on this, guess the instantaneous velocity at t=1

Answer 1

Hi Morgen,

Here are the answers:

For 0.1 seconds, the average velocity is 36.4 ft/sec.

For 0.01 seconds, the average velocity is 37.84 ft/sec.

For 0.001 seconds, the average velocity is 37.984 ft/sec.

Instantaneous velocity at t=1 is 38 ft/sec.

Below is an explanation of how I arrived at the above answers:

We define average velocity as the change in displacement divided by the change in time. To calculate the change in the displacement, we subtract the starting position of the ball from the ending position of the ball. To calculate the change in time, we subtract the start time from the end time. In symbols, the equation for average velocity is:

v_{avg =} (x_end - x_start) / (t_end - t_start)

The position of the ball as a function of time, x(t), with an initial velocity of 70 ft/sec is:

x(t) = 70t - 16t²

Now, we have all of the information we need to calculate our average velocities. Let's consider the time interval of 0.1 seconds.

x_end = x(1.1) = 70*(1.1)-16*(1.1)² = 57.64

x_start = x(1) = 70*1-16*(1)² = 54

Therefore, the average velocity is:

v_avg = (57.64-54) / (1.1 - 1) = 36.4 ft/sec

We use similar logic to calculate the average velocity over 0.01 and 0.001 seconds.

The average velocity is approaching 38 ft/sec as the length of the time interval approaches 0, so the instantaneous velocity at t=1 is 38 ft/sec.

I hope this helps.

Answer 2

First, you'd need to figure out what the definition of average velocity is in terms of the function of height. We know that velocity is a derivative of height, and we can use the limit definition of derivative:

For example, if we were to find the average velocity beginning at t=1 and lasting .2 seconds, we would have

(f(1.2)-f(1))/(1.2-1), where f(t) is the height function.

Hope this helps!