To solve this problem, we can use the equations of motion and the principles of projectile motion. Let's break down the problem step by step:
(a) To find the minimum initial velocity, we need to determine the speed at which the ball leaves the racquet. We can use the equation:
v^2 = u^2 + 2as
where:
v = final velocity (0 m/s since the ball reaches maximum height)
u = initial velocity (what we're trying to find)
a = acceleration (due to gravity, approximately -9.8 m/s^2)
s = displacement (vertical distance, which is the difference in height)
Given the displacement s = 2.40 m - 0.900 m + 0.200 m = 1.700 m, and since the ball reaches its maximum height (v = 0 m/s), we can rearrange the equation to solve for u:
u^2 = v^2 - 2as
u^2 = 0^2 - 2(-9.8 m/s^2)(1.7 m)
u^2 = 33.32 m^2/s^2
u = √(33.32 m^2/s^2)
u ≈ 5.77 m/s
Therefore, the minimum initial velocity needed for the tennis ball to clear the net by at least 0.200 m is approximately 5.77 m/s.
(b) To find where the ball will land, we need to determine the horizontal distance it travels. We can use the equation:
d = ut + 0.5at^2
where:
d = horizontal distance (what we're trying to find)
u = initial velocity (5.77 m/s)
a = acceleration (0 m/s^2 since there is no horizontal acceleration)
t = time of flight (what we're trying to find)
Since the ball is served at an angle above the horizontal, we can decompose the initial velocity into horizontal and vertical components:
u_x = u * cosθ
u_y = u * sinθ
where:
θ = angle above the horizontal (2.00 degrees in this case)
Given u = 5.77 m/s and θ = 2.00 degrees, we can calculate the horizontal and vertical initial velocities:
u_x = 5.77 m/s * cos(2.00 degrees)
u_y = 5.77 m/s * sin(2.00 degrees)
Now we can determine the time of flight using the vertical motion equation:
y = u_y * t + 0.5 * a * t^2
where:
y = vertical displacement (height of the net, which is 0.900 m)
Plugging in the values:
0.900 m = (5.77 m/s * sin(2.00 degrees)) * t + 0.5 * (-9.8 m/s^2) * t^2
This is a quadratic equation in t, which can be solved to find the time of flight (t).
Once we have the time of flight, we can calculate the horizontal distance using:
d = u_x * t
Therefore, by calculating the time of flight and multiplying it by the horizontal component of the velocity, we can determine where the ball will land.