A tennis ball is served from the back line of the court such that it leaves the racket 2.4 m above the ground in a horizontal direction at a speed of 22.3 m/s. (a) will the ball cross a 0.91-m-high net in front of the server? Will the ball land in the service court, which is within 6.4 m if the net on the other side of the net
h = 0.5g*t^2 = 2.4-0.91.
4.9t^2 = 1.49. t1 = 0.551 s.
d1 = Xo*t1 = 22.3 * 0.551 = 12.3 m. = Distance from back line to net.
0.5g*t2^2 = 2.4.
4.9t2^2 = 2.4, t2 = 0.70 s. to fall to gnd.
d2 = Xo*t2 = 22.3 * 0.7 = 15.6 m. = Maximum distance travelled.
a. Yes.
b. d = 12.3 + 6.4 = 18.7 m. available. Yes.
To determine if the tennis ball will cross the net and land within the service court, we need to analyze its projectile motion.
First, let's find the time it takes for the ball to reach the net using the horizontal velocity:
Using the equation:
Distance = Velocity × Time
We can rearrange it to solve for time:
Time = Distance / Velocity
Time = 6.4 m / 22.3 m/s
Time = 0.287 seconds (approximately)
Now, let's determine the vertical displacement of the ball when it reaches the net. We can use the equation for vertical displacement in projectile motion:
Vertical Displacement = Initial Vertical Velocity × Time + (1/2) × Acceleration × Time²
Given:
Initial Vertical Velocity = 0 m/s (since the ball is leaving the racket horizontally)
Acceleration = -9.8 m/s² (due to gravity)
Time = 0.287 seconds
Vertical Displacement = 0 × 0.287 + (1/2) × (-9.8) × (0.287)²
Vertical Displacement ≈ -0.4256 meters
The negative sign indicates that the ball falls below the net height.
Since the ball's vertical displacement is -0.4256 meters, which is lower than the 0.91-meter net height, the ball will cross the net.
Now, let's determine if the ball will land within the service court. To do this, we'll find the horizontal distance the ball travels using its time of flight:
Using the equation:
Distance = Velocity × Time
Distance = 22.3 m/s × 0.287 seconds
Distance ≈ 6.4 meters
Since the ball travels a distance of approximately 6.4 meters horizontally within the given time, it will land within the service court, which is also within 6.4 meters on the opposite side of the net.
Therefore, the ball will cross the net and land within the service court.
To determine whether the tennis ball will cross the net and land within the service court, we can analyze the motion of the ball using projectile motion equations. Here's how you can solve this problem step by step:
Step 1: Identify the given variables and known values:
- The vertical position of the racket above the ground (h) = 2.4 m
- The horizontal velocity of the ball (Vx) = 22.3 m/s
- The height of the net (H_net) = 0.91 m
- The width of the service court (D_court) = 6.4 m
Step 2: Analyze the vertical motion:
Since the ball is only affected by gravity vertically, we need to determine the vertical distance the ball travels.
Using the formula for vertical displacement (Δy) in projectile motion:
Δy = Vyi * t + (1/2) * (-g) * t^2
Since the ball is served horizontally, the initial vertical velocity (Vyi) is 0. The only acceleration acting on the ball is due to gravity (g = 9.8 m/s^2). Therefore, the equation simplifies to:
Δy = (1/2) * (-g) * t^2
Step 3: Find the time of flight:
To find the time it takes for the ball to reach the other side, we can use the horizontal velocity and the distance (D_court) between the server's position and the other side.
Using the formula for horizontal distance (Δx) in projectile motion:
Δx = Vxi * t
where Vxi is the initial horizontal velocity, which remains constant throughout the motion.
Δx = Vx * t
Since Δx = D_court, we can find t as follows:
t = D_court / Vx
Step 4: Calculate the vertical displacement:
Now, substitute the value of t (found in Step 3) into the equation for vertical displacement (Step 2).
Δy = (1/2) * (-g) * (D_court / Vx)^2
Step 5: Analyze the result:
If the vertical displacement (Δy) exceeds the height of the net (H_net), then the ball will cross the net. If Δy is also less than or equal to the height of the net (H_net), the ball will land within the service court.
Calculate the values and draw your conclusions based on the results obtained.