A tennis player standing 11.0 m from the net hits the ball at 3.34° above the horizontal. To clear the net, the ball must rise at least 0.320 m. If the ball just clears the net at the apex of its trajectory, how fast was the ball moving when it left the racquet?

To find the initial velocity of the ball when it left the racquet, we can use projectile motion equations.

First, let's break down the given information:
- The distance from the player to the net is 11.0 m.
- The ball must rise at least 0.320 m to clear the net.
- The angle above the horizontal at which the ball is hit is 3.34°.

Now, let's find the initial velocity (v₀) of the ball.

Step 1: Find the vertical component of the initial velocity.
The vertical component (v_y) can be calculated using the formula:
v_y = v₀ * sinθ

where θ is the given angle above the horizontal.

v_y = v₀ * sin(3.34°)

Step 2: Find the time of flight (t) to reach the apex of the trajectory.
The time of flight is the time taken by the ball to reach the highest point of its trajectory. At this point, the vertical displacement is half of the total vertical distance covered.

Using the formula:
h = v_y * t - (1/2) * g * t²

where h is the total vertical distance covered (0.320 m), g is the acceleration due to gravity (approximately 9.8 m/s²), and t is the time of flight.

0.320 = (v₀ * sin(3.34°)) * t - (1/2) * (9.8) * t²

Step 3: Find the horizontal component of the initial velocity.
The horizontal component (v_x) can be calculated using the formula:
v_x = v₀ * cosθ

where θ is the given angle above the horizontal.

v_x = v₀ * cos(3.34°)

Step 4: Calculate the time taken to cover the horizontal distance.
The horizontal distance covered (d) can be calculated using the formula:
d = v_x * t

where d is the given distance from the player to the net (11.0 m), v_x is the horizontal component of the initial velocity, and t is the time taken to cover the horizontal distance.

11.0 = (v₀ * cos(3.34°)) * t

Step 5: Solve the system of equations.
Combine the equations from Step 2 and Step 4 to solve for v₀.
We have two equations with two variables (v₀ and t). Solve these equations simultaneously to find the value of v₀.

Now that we have v₀, the initial velocity of the ball when it left the racquet, we have the answer to the question.