A ball is thrown straight up with an initial velocity of 224 ft/sec, so its height (in feet) is given by h(t)=224t-16t². b. Find the average velocity of the ball over the interval [4,4.1] c. When will the ball hit the ground? d. Find the inverse function that gives the time t at which the ball is at height h as the ball travels downward.
A ball is thrown straight up with an initial velocity of 224 ft/sec, so its height (in feet) is given by h(t)=224t-16t². b. Find the average velocity of the ball over the interval [4,4.1] c. When will the ball hit the ground? d. Find the inverse function that gives the time t at which the ball is at height h as the ball travels downward.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![The problem involves the motion of a ball thrown straight upward with an initial velocity of 224 ft/sec. The height of the ball at time \( t \) is given by the function:
\[ h(t) = 224t - 16t^2 \]
**Questions:**
a. (Not specified.)
b. Find the average velocity of the ball over the interval \([4, 4.1]\).
c. Determine when the ball will hit the ground.
d. Find the inverse function that provides the time \( t \) at which the ball is at a height \( h \) while traveling downward.
e. Find the time at which the ball will be descending and at a height of 720 feet above the ground.
f. Create a table of values and record the average velocity in the intervals shortly before the ball hits the ground. Assume the answer to when it hits the ground is \( b \).
**Table:**
| Time Frame | Time Interval | Average Velocity |
|----------------------------------------|-------------------|------------------|
| 1 second before hitting ground | | |
| ½ second before hitting ground | | |
| 0.1 second before hitting ground | | |
| 0.01 seconds before hitting ground | | |
| 0.001 seconds before hitting ground | | |
g. Estimate the speed of the ball immediately before it hits the ground.
**Explanation:**
- The function \( h(t) = 224t - 16t^2 \) represents the height in feet of the ball at any time \( t \).
- The average velocity over an interval can be calculated using the change in height divided by the change in time.
- When the ball hits the ground, \( h(t) = 0 \).
- The inverse function would allow calculation of \( t \) given \( h \) for the downward path.
- Understanding these concepts is critical in analyzing the motion of objects in physics, specifically under the influence of gravity.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc84d0291-1107-4711-af5a-30afb213fbbf%2F33b637da-3f9b-42b7-a612-48744cb817c4%2Fwxerdpn_processed.png&w=3840&q=75)
Transcribed Image Text:The problem involves the motion of a ball thrown straight upward with an initial velocity of 224 ft/sec. The height of the ball at time \( t \) is given by the function:
\[ h(t) = 224t - 16t^2 \]
**Questions:**
a. (Not specified.)
b. Find the average velocity of the ball over the interval \([4, 4.1]\).
c. Determine when the ball will hit the ground.
d. Find the inverse function that provides the time \( t \) at which the ball is at a height \( h \) while traveling downward.
e. Find the time at which the ball will be descending and at a height of 720 feet above the ground.
f. Create a table of values and record the average velocity in the intervals shortly before the ball hits the ground. Assume the answer to when it hits the ground is \( b \).
**Table:**
| Time Frame | Time Interval | Average Velocity |
|----------------------------------------|-------------------|------------------|
| 1 second before hitting ground | | |
| ½ second before hitting ground | | |
| 0.1 second before hitting ground | | |
| 0.01 seconds before hitting ground | | |
| 0.001 seconds before hitting ground | | |
g. Estimate the speed of the ball immediately before it hits the ground.
**Explanation:**
- The function \( h(t) = 224t - 16t^2 \) represents the height in feet of the ball at any time \( t \).
- The average velocity over an interval can be calculated using the change in height divided by the change in time.
- When the ball hits the ground, \( h(t) = 0 \).
- The inverse function would allow calculation of \( t \) given \( h \) for the downward path.
- Understanding these concepts is critical in analyzing the motion of objects in physics, specifically under the influence of gravity.
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