If a ball is thrown straight up into the air with an initial velocity of 70 ft/s, its height in feet after t
 second is given by y=70t−16t2
. Find the average velocity (include units, ) for the time period beginning when t=8
 seconds and lasting
Part 1 : 1 point(s)
(i) 0.8 seconds
The average velocity after lasting 0.8 seconds is 
 ft/s.

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Part 2 : 1 point(s)
(ii) 0.2 seconds
After lasting for 0.2 seconds, the average velocity is 
 ft/s.

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Part 3 : 1 point(s)
(iii) 0.02 seconds
The average velocity after lasting 0.02 seconds is 
 ft/s.
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Part 4 : 1 point(s)
(iv) Finally based on the above results, guess what the instantaneous velocity of the ball is when t=8
. 
The instantaneous velocity of the ball is 
 ft/s.

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To solve for the average velocity over each time interval and estimate the instantaneous velocity of the ball at \( t = 8 \) seconds, we will use the following concepts:

1. The height of the ball at any time \( t \) is given by:
   \[
   y(t) = 70t - 16t^2
   \]
2. The formula for average velocity over a time interval \( [t_1, t_2] \) is:
   \[
   \text{Average velocity} = \frac{y(t_2) - y(t_1)}{t_2 - t_1}
   \]
   where \( t_1 \) is the starting time and \( t_2 \) is the ending time.

For each part of the problem, we are asked to calculate the average velocity over a short time period starting at \( t_1 = 8 \) seconds.

### Part 1: Time period lasting 0.8 seconds

- Initial time: \( t_1 = 8 \) seconds
- Final time: \( t_2 = 8 + 0.8 = 8.8 \) seconds

We need to find the heights at these two times, \( y(8) \) and \( y(8.8) \), and then calculate the average velocity:
\[
\text{Average velocity} = \frac{y(8.8) - y(8)}{8.8 - 8}
\]

### Part 2: Time period lasting 0.2 seconds

- Initial time: \( t_1 = 8 \) seconds
- Final time: \( t_2 = 8 + 0.2 = 8.2 \) seconds

The same process applies, but now for \( t_2 = 8.2 \):
\[
\text{Average velocity} = \frac{y(8.2) - y(8)}{8.2 - 8}
\]

### Part 3: Time period lasting 0.02 seconds

- Initial time: \( t_1 = 8 \) seconds
- Final time: \( t_2 = 8 + 0.02 = 8.02 \) seconds

We calculate the average velocity over this very small time interval:
\[
\text{Average velocity} = \frac{y(8.02) - y(8)}{8.02 - 8}
\]

### Part 4: Estimate the instantaneous velocity at \( t = 8 \)

The instantaneous velocity is the derivative of the height function \( y(t) = 70t - 16t^2 \) with respect to \( t \), evaluated at \( t = 8 \). The derivative represents the velocity function:
\[
v(t) = \frac{dy}{dt} = 70 - 32t
\]
Substituting \( t = 8 \) to find the instantaneous velocity:
\[
v(8) = 70 - 32(8) = 70 - 256 = -186 \, \text{ft/s}
\]

I'll now calculate the average velocities for parts 1-3.Here are the results for each part:

### Part 1:
The average velocity over the 0.8 second interval is approximately **-198.8 ft/s**.

### Part 2:
The average velocity over the 0.2 second interval is approximately **-189.2 ft/s**.

### Part 3:
The average velocity over the 0.02 second interval is approximately **-186.32 ft/s**.

### Part 4:
The instantaneous velocity at \( t = 8 \) seconds is estimated to be **-186 ft/s**.

Would you like more details on any of the steps or have any other questions?

Here are five related questions to deepen your understanding:
1. How does the ball's height change over time, and what does this indicate about its motion?
2. Can you explain the significance of the negative sign in the velocity?
3. How is average velocity different from instantaneous velocity?
4. What would happen to the velocity if the initial velocity were increased?
5. How would you calculate the velocity at \( t = 0 \)?

**Tip:** The instantaneous velocity is the slope of the tangent line to the height curve at a specific point in time.

If a ball is thrown straight up into the air with an initial velocity of 70 ft/s, its height in feet after t seconds is given by y=70t−16t^2. Find the average velocity for the time period beginning when t=8 seconds and lasting (i) 0.8 seconds, (ii) 0.2 seconds, (iii) 0.02 seconds. Finally, estimate the instantaneous velocity at t=8 seconds.

Learn how to calculate the average and instantaneous velocity of a ball thrown upward with an initial velocity of 70 ft/s. This problem involves the kinematic equation for height and explores key concepts in derivatives and average velocity. Suitable for students in Grades 10-12.

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