To determine the speed of the ball as it is released, we can use kinematic equations. When a ball is thrown vertically upward and returns to its starting point, the time to reach the highest point is half of the total time, as the upward and downward journeys are symmetric.

Given dataTo determine the speed of the ball as it is released, we can use kinematic equations. When a ball is thrown vertically upward and returns to its starting point, the time to reach the highest point is half of the total time, as the upward and downward journeys are symmetric.

Given

• Total time for the ball to return to the point of release, $t_{\text{total}} = 3.58 \, \text{s}$
• Acceleration due to gravity, $g = 9.8 \, \text{m/s}^2$

Solution:

1. Calculate the time to reach the highest point: $t_{\text{up}} = \frac{t_{\text{total}}}{2} = \frac{3.58 \, \text{s}}{2} = 1.79 \, \text{s}$

2. Use the kinematic equation for velocity: At the highest point, the velocity of the ball is zero. The initial velocity $v_0$ needed to reach this point can be calculated using: $v = v_0 - g \cdot t_{\text{up}}$ where $v = 0$ (since the ball momentarily stops at the top).

   Rearranging for $v_0$: $v_0 = g \cdot t_{\text{up}}$

3. Substitute the values: $v_0 = 9.8 \, \text{m/s}^2 \times 1.79 \, \text{s} = 17.542 \, \text{m/s}$

Answer:

The speed of the ball as it is released is approximately: $v_0 \approx 17.54 \, \text{m/s}$

Would you like further details, or do you have any questions?

Here are some related questions to explore:

1. How would the release speed change if the ball were thrown at an angle instead of vertically?
2. What would be the maximum height reached by the ball in this scenario?
3. How would air resistance affect the motion and calculations of the ball?
4. What if the gravitational constant $g$ was different? How would that impact the release speed?
5. How can we determine the time the ball spends at each specific height during its flight?

Tip: For projectile motion problems, always break down the motion into vertical and horizontal components, even if one of them is zero (as in vertical motion).