The lob in tennis is an effective tactic when your opponent is near the net. It consists of lofting the ball over his head, forcing him to move quickly away from the net (as shown in the figure below).
Suppose that you loft the ball with an initial speed of v = 15.6 m/s, at an angle of θ = 48.1° above the horizontal. At this instant your opponent is d = 11.4 m away from the ball. He begins moving away from you 0.350 s later, hoping to reach the ball and hit it back at the moment that it is h = 2.18 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.)
Well, it sounds like your opponent is in a bit of a pickle here. Let's see if we can help him out with a little bit of clownish calculations.
To find the minimum average speed your opponent needs to reach the ball, we need to consider a few things. First, we need to figure out how long it takes for the ball to reach its maximum height, h = 2.18 m.
The time it takes for the ball to reach its maximum height can be found using the formula:
t = (v * sin(θ)) / g
where v is the initial speed of the ball, θ is the launch angle, and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Plugging in the given values, we have:
t = (15.6 m/s * sin(48.1°)) / 9.8 m/s^2
Now that we have the time it takes for the ball to reach its maximum height, we can calculate how far your opponent needs to move during this time. This can be found using the formula:
d = v * cos(θ) * t
where d is the initial distance between your opponent and the ball.
Plugging in the values, we have:
11.4 m = 15.6 m/s * cos(48.1°) * t
Now, solving for t, we find:
t = 11.4 m / (15.6 m/s * cos(48.1°))
Next, we need to calculate how fast your opponent needs to move in order to cover this distance in the given time.
The average speed is given by the formula:
average speed = total distance / total time
In this case, the total distance is the distance D that your opponent needs to cover, which is the initial distance d plus the distance the ball travels during time t:
D = d + v * sin(θ) * t
Plugging in the values, we have:
D = 11.4 m + 15.6 m/s * sin(48.1°) * t
Finally, we can substitute the expression for t that we found earlier into the equation for D:
D = 11.4 m + 15.6 m/s * sin(48.1°) * (11.4 m / (15.6 m/s * cos(48.1°)))
And now, after juggling all these numbers, we find the minimum average speed your opponent must move with a flourish:
average speed = D / t
So go ahead, plug in the numbers and give it a whirl!
To answer this question, we can use the equations of projectile motion. The key is to find the time it takes for the ball to reach the maximum height (h = 2.18 m) and the time it takes for the ball to travel horizontally to reach the opponent's initial position (d = 11.4 m). Once we have these times, we can calculate the minimum average speed the opponent needs to move.
1. Find the time to reach maximum height:
The vertical motion of the ball is given by the equation:
h = (v₀ * sin(θ) * t) - (0.5 * g * t²), where v₀ is initial velocity, θ is the angle, t is time, and g is gravity (9.8 m/s²).
At the maximum height, the vertical velocity is zero (vₓ = 0).
Solve for time by setting vₓ = 0:
0 = v₀ * sin(θ) - g * t
t = v₀ * sin(θ) / g
2. Find the time to reach the opponent's initial position:
The horizontal motion of the ball is given by the equation:
d = v₀ * cos(θ) * t
Solve for time:
t = d / (v₀ * cos(θ))
3. Calculate the minimum average speed of the opponent:
The opponent needs to reach the ball by the time it reaches the maximum height. Since the opponent starts moving 0.350 s later, the time available for the opponent to reach the ball is:
t_available = t_max_height - t_opponent_delay
t_available = v₀ * sin(θ) / g - 0.350
The distance the opponent needs to cover horizontally in this time is the horizontal distance the ball travels in the same time period:
d_opponent = v₀ * cos(θ) * t_available
To find the minimum average speed, divide the total distance traveled by the time taken:
minimum average speed = d_opponent / t_available
Now, you can substitute the given values (v₀ = 15.6 m/s, θ = 48.1°, d = 11.4 m, h = 2.18 m, and g = 9.8 m/s²) into the equations and calculate the minimum average speed.