A ball reaches a maximum height of h = 86 meters.
when thrown vertically upwards. Assume gravitational acceleration g = -9.8m/s^2.



What was the initial velocity of the ball?
 
Enter answer in meters and in three significant figures.

We can use the following kinematic equation to solve for the initial velocity \(v_0\):

\[
v^2 = v_0^2 + 2a h
\]

Where:
- \(v\) is the final velocity at maximum height (which is 0 because the ball stops momentarily at its highest point),
- \(v_0\) is the initial velocity (what we're solving for),
- \(a\) is the acceleration due to gravity, \(a = -9.8 \, \text{m/s}^2\),
- \(h\) is the maximum height, \(h = 86 \, \text{m}\).

At the maximum height, \(v = 0\), so the equation simplifies to:

\[
0 = v_0^2 + 2(-9.8)(86)
\]

Now, solving for \(v_0\):

\[
v_0^2 = -2(-9.8)(86)
\]
\[
v_0^2 = 1685.6
\]
\[
v_0 = \sqrt{1685.6} \approx 41.04 \, \text{m/s}
\]

Therefore, the initial velocity of the ball is approximately \( \mathbf{41.0 \, \text{m/s}} \) (to three significant figures).

Would you like further details or have any questions?

Here are five related questions you might consider:
1. How long does it take the ball to reach its maximum height?
2. What is the total time of flight of the ball?
3. How fast is the ball moving when it returns to the ground?
4. What is the displacement of the ball after a certain time interval during its upward motion?
5. How would air resistance affect the motion of the ball?

**Tip:** The kinematic equations are very useful when an object moves with constant acceleration, like in free-fall problems.

A ball reaches a maximum height of h = 86 meters when thrown vertically upwards. Assume gravitational acceleration g = -9.8m/s^2. What was the initial velocity of the ball? Enter the answer in meters and in three significant figures.

Learn how to calculate the initial velocity of a ball thrown upwards to a maximum height of 86 meters using kinematic equations and gravitational acceleration. This problem is suitable for Grades 9-12 and involves core concepts of projectile motion and kinematics.

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