Question
What would be the initial velocity required for the ball to reach a height of 10 feet?
Answers
GPT-4o mini
To determine the initial velocity required for a ball to reach a height of 10 feet, we can use the kinematic equations of motion under the influence of gravity. The relevant equation for vertical motion is:
\[
v^2 = u^2 + 2as
\]
where:
- \( v \) is the final velocity (0 ft/s at the peak height),
- \( u \) is the initial velocity (what we are trying to find),
- \( a \) is the acceleration (for upward motion against gravity, this will be -32.2 ft/s², the approximate acceleration due to gravity),
- \( s \) is the displacement (the height we want to achieve, which is 10 feet).
At the peak height (10 feet), the final velocity \( v = 0 \). We can now rearrange the equation to solve for \( u \):
\[
0 = u^2 + 2(-32.2)(10)
\]
Simplifying:
\[
0 = u^2 - 644
\]
\[
u^2 = 644
\]
\[
u = \sqrt{644} \approx 25.38 \text{ ft/s}
\]
Thus, the initial velocity required for the ball to reach a height of 10 feet is approximately **25.38 feet per second**.
\[
v^2 = u^2 + 2as
\]
where:
- \( v \) is the final velocity (0 ft/s at the peak height),
- \( u \) is the initial velocity (what we are trying to find),
- \( a \) is the acceleration (for upward motion against gravity, this will be -32.2 ft/s², the approximate acceleration due to gravity),
- \( s \) is the displacement (the height we want to achieve, which is 10 feet).
At the peak height (10 feet), the final velocity \( v = 0 \). We can now rearrange the equation to solve for \( u \):
\[
0 = u^2 + 2(-32.2)(10)
\]
Simplifying:
\[
0 = u^2 - 644
\]
\[
u^2 = 644
\]
\[
u = \sqrt{644} \approx 25.38 \text{ ft/s}
\]
Thus, the initial velocity required for the ball to reach a height of 10 feet is approximately **25.38 feet per second**.