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(1 point) A regulation volleyball court is L = 18.0 m long and a regulation volleyball net is d = 2.43 m high. A volleyball player strikes the ball a height h = 1.77 m directly above the back line, and the ball's initial velocity makes an angle theta = 51o with respect to the ground (see the figure). At what initial speed must the ball be hit so that it just barely makes it over the net? (Assume the volleyball is hit so that its path is parallel to the side-line as seen from an observer directly above the court, and that the volleyball is a point object.)

We are tasked with calculating the initial speed \( v_0 \) at which the volleyball must be hit so that it just barely clears the net, given the following:

- The length of the volleyball court \( L = 18.0 \, \text{m} \),
- The height of the net \( d = 2.43 \, \text{m} \),
- The height from which the volleyball is hit \( h = 1.77 \, \text{m} \),
- The angle of launch \( \theta = 51^\circ \).

We will use the standard projectile motion equations to solve this. The two key components of motion are:
1. Horizontal motion: \( x = v_0 \cos(\theta) \cdot t \),
2. Vertical motion: \( y = h + v_0 \sin(\theta) \cdot t - \frac{1}{2} g t^2 \),

where \( g \) is the acceleration due to gravity, \( g \approx 9.8 \, \text{m/s}^2 \), and \( t \) is the time the ball takes to travel.

We are particularly interested in the point when the ball reaches the net. The horizontal distance to the net is half the court length, i.e., \( x = \frac{L}{2} = 9.0 \, \text{m} \), and the vertical position of the ball at that point must be equal to the height of the net, i.e., \( y = d = 2.43 \, \text{m} \).

Let me now solve this equation for \( v_0 \).The required initial speed at which the ball must be hit to just barely clear the net is approximately \( v_0 = 9.79 \, \text{m/s} \). The negative value is physically irrelevant, so the positive value is the correct solution.

Would you like a detailed explanation of the steps involved or further questions related to this?

Here are some related questions to extend this topic:
1. What would be the time of flight for the ball to reach the net?
2. How would the initial speed change if the height of the net were different?
3. What is the maximum height the ball reaches during its flight?
4. How would air resistance affect the required initial speed?
5. What would be the distance traveled if the launch angle were increased or decreased slightly?

**Tip:** The horizontal and vertical components of motion are independent of each other, but they must satisfy the conditions of the entire motion trajectory simultaneously.

A regulation volleyball court is L = 18.0 m long and a regulation volleyball net is d = 2.43 m high. A volleyball player strikes the ball a height h = 1.77 m directly above the back line, and the ball's initial velocity makes an angle θ = 51° with respect to the ground. At what initial speed must the ball be hit so that it just barely makes it over the net?

Calculate the initial speed required for a volleyball hit at an angle of 51° to just clear a net 2.43 m high on a regulation volleyball court 18 m long. Using projectile motion equations and kinematics, we determine the necessary velocity for this trajectory.

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