A tennis player standing 9.7 m from the net hits the ball at 3.42° above the horizontal. To clear the net, the ball must rise at least 0.291 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?
Y^2 = Yo^2 + 2g*h = 0
Yo^2 - 19.6*0.291 = 0
Yo^2 = 5.70
Yo = 2.39 m/s = Ver. component of initial velocity.
Vo = Yo/sin A = 2.39/sin3.42 = 40.0 m/s.
[3.42o]
To find the initial velocity of the ball when it left the racquet, we can use the principles of projectile motion and kinematics.
We know the following:
1. The horizontal distance from the player to the net (range): 9.7 m
2. The vertical displacement of the ball (maximum height): 0.291 m
3. The launch angle of the ball above the horizontal: 3.42°
Using these values, we can first calculate the time it takes for the ball to reach its maximum height (apex) using the vertical motion equation:
h = (viy * t) + (0.5 * g * t^2),
where
- h is the vertical displacement (0.291 m),
- viy is the vertical component of the initial velocity (unknown),
- t is the time it takes to reach the maximum height, and
- g is the acceleration due to gravity (9.8 m/s^2).
Since the ball reaches its maximum height, the final vertical velocity at the apex is 0 m/s. Therefore, the equation becomes:
0 = (viy * t) + (0.5 * g * t^2).
Next, we need to find the time it takes for the ball to reach the net. We can use the horizontal motion equation:
x = vix * t,
where
- x is the horizontal distance to the net (9.7 m),
- vix is the horizontal component of the initial velocity (unknown), and
- t is the time it takes to reach the net (same as the time to reach the apex).
Now, we have two equations with two unknowns (viy and vix). We can solve for t by dividing the second equation by the first equation:
(0.5 * g * t^2) = (x / t).
Simplifying, we get:
0.5 * g * t^2 - (x / t) = 0.
Now, we can solve this quadratic equation for t. Plugging in the known values of x and g, we get:
0.5 * 9.8 * t^2 - (9.7 m / t) = 0.
Solving this equation for t will give us the time it takes for the ball to reach the net. Once we have t, we can substitute it back into either of the original equations to solve for viy.
Finally, we can use the launch angle and the viy component to find the initial velocity of the ball. Using trigonometry, we have:
vi = viy / sin(θ),
where
- vi is the initial velocity of the ball leaving the racquet, and
- θ is the launch angle (3.42°).
Solving for vi will give us the answer to the question: the speed of the ball when it left the racquet.