You serve a tennis ball from a height of 2.29 m above the ground. The ball leaves your racket with a speed of 18.7 m/s at an angle of 6.55° above the horizontal. The horizontal distance from the court's baseline to the net is 11.89 m, and the net is 1.07 m high. Neglect spin imparted on the ball as well as air resistance effects. Does the ball clear the net (= positive answer)? If yes, by how much? If not, by how much did it miss? In that case the answer will be negative.
To determine if the tennis ball clears the net, we need to calculate its vertical position when it reaches the net. If the ball's vertical position is higher than the net's height, it clears the net. Otherwise, it does not clear the net.
First, let's calculate the time it takes for the ball to reach the net. We can use the vertical component of the initial velocity and the acceleration due to gravity (9.8 m/s^2) to find the time of flight.
The initial vertical velocity (Vy0) is given by:
Vy0 = velocity * sin(angle)
Vy0 = 18.7 m/s * sin(6.55°)
Vy0 ≈ 18.7 m/s * 0.1143
Vy0 ≈ 2.137 m/s
Next, we can use the equation for vertical displacement to find the time of flight:
Δy = Vy0 * t + (1/2) * a * t^2
Where:
Δy = vertical displacement (net height)
Vy0 = initial vertical velocity
a = acceleration due to gravity (-9.8 m/s^2)
t = time of flight
Since Δy is the height of the net (-1.07 m), we can rearrange the equation to solve for t:
-1.07 m = 2.137 m/s * t + (1/2) * (-9.8 m/s^2) * t^2
This equation is a quadratic equation. Solving it will give us the time at which the ball reaches the net, and we can determine if it clears the net or not.
Now, we have the following quadratic equation:
(1/2) * (-9.8 m/s^2) * t^2 + 2.137 m/s * t - 1.07 m = 0
To solve this equation, we can use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = (1/2) * (-9.8 m/s^2), b = 2.137 m/s, and c = -1.07 m. Substituting these values into the quadratic formula will give us the time of flight.
t = (-2.137 m/s ± √((2.137 m/s)^2 - 4 * (1/2) * (-9.8 m/s^2) * (-1.07 m))) / (2 * (1/2) * (-9.8 m/s^2))
Simplifying this equation will give us two values for t, as it's quadratic. We only need the positive value since time cannot be negative in this context.
By solving the equation, the positive value of t is approximately 0.256 seconds.
Now, we can calculate the vertical position of the ball at this time:
y = (1/2) * a * t^2
Where:
y = vertical displacement (negative if below the net, positive if above)
a = acceleration due to gravity (-9.8 m/s^2)
t = time of flight
Using these values, we can find the vertical position of the ball when it reaches the net:
y = (1/2) * (-9.8 m/s^2) * (0.256 s)^2
By calculating the value, we find that y ≈ -0.307 m, which is below the net. Therefore, the ball does not clear the net.
To determine how much the ball missed by, we can calculate the difference between the net's height and the ball's vertical position at the net:
miss = net height - vertical position at the net
miss = 1.07 m - (-0.307 m)
miss ≈ 1.377 m
Therefore, the ball misses the net by approximately 1.377 meters.