Suppose that you lob the ball with an initial speed of v = 16.1 m/s, at an angle of θ = 51.9° above the horizontal. At this instant your opponent is d = 11.2 m away from the ball. He begins moving away from you 0.400 s later, hoping to reach the ball and hit it back at the moment that it is h = 2.06 m above its launch point. With what minimum average speed must he move? (Ignore the fact that he can stretch, so that his racket can reach the ball before he does.)
To find the minimum average speed at which the opponent must move in order to reach the ball, we need to consider the horizontal and vertical components of motion separately.
Let's start with the horizontal component. We know the initial speed of the ball is 16.1 m/s and the opponent is 11.2 m away from the ball. We also know that the opponent starts moving 0.400 s after the ball is launched. This means the opponent has (0.400 s) + (time taken by the ball to reach the opponent) to cover the 11.2 m distance.
To find the time taken by the ball to reach the opponent, we can use the horizontal component of its initial velocity. We can find this using the formula:
Vx = V * cos(θ)
where Vx is the horizontal component of the initial velocity, V is the initial velocity of the ball, and θ is the launch angle.
Substituting the given values:
Vx = 16.1 m/s * cos(51.9°)
Now, we can find the time taken by the ball to reach the opponent using the formula:
time = distance / velocity
where distance is the 11.2 m distance and velocity is the horizontal component of the initial velocity.
time = 11.2 m / (16.1 m/s * cos(51.9°))
Now that we know the time taken by the ball to reach the opponent, we can calculate how much time the opponent has to cover the remaining distance (11.2 m) to reach the ball. This is the time remaining after subtracting the time taken by the ball to reach the opponent from the total time (0.400 s + time taken by the ball), which is:
remaining time = (0.400 s + time taken by the ball) - time taken by the ball to reach the opponent
Now that we have the remaining time, we can calculate the average speed at which the opponent must move to cover the remaining distance of 11.2 m:
average speed = remaining distance / remaining time
average speed = 11.2 m / remaining time
The remaining time is the time taken by the ball plus the additional time the opponent has to cover the remaining distance.
Finally, we can substitute the values calculated in the above steps to find the minimum average speed at which the opponent must move.