A tennis player serves a ball horizontally at a height h = 3.4 m a distance d = 15.3 m from the net. What maximum speed can the ball be served so as to land within a distance w = 6.3 m of the and be a good serve?
To find the maximum speed at which the ball can be served, we need to first calculate the time it takes for the ball to reach the net.
We can use the kinematic equation:
h = (1/2)gt^2
where h is the height, g is the acceleration due to gravity (approximately 9.8 m/s^2), and t is the time.
Rearranging the equation:
t = √(2h/g)
Plugging in the values:
t = √(2 * 3.4 / 9.8)
t ≈ √(0.693877551)
Now, we can calculate the horizontal velocity (v) using the formula:
d = vt
where d is the distance and t is the time of flight.
Rearranging the equation:
v = d / t
Plugging in the values:
v = 15.3 / √(0.693877551)
v ≈ 15.3 / 0.833333333
Finally, we calculate the maximum speed (v_max) by subtracting the desired landing distance (w) from the horizontal velocity (v):
v_max = v - w
Plugging in the values:
v_max ≈ (15.3 / 0.833333333) - 6.3
Calculating the result will give you the maximum speed at which the ball can be served to meet the given conditions.