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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 (see the drawing). Suppose that you lob the ball with an initial speed of \(15.0 \mathrm{m} / \mathrm{s},\) at an angle of \(50.0^{\circ}\) above the horizontal. At this instant your opponent is \(10.0 \mathrm{m}\) away from the ball. He begins moving away from you 0.30 s later, hoping to reach the ball and hit it back at the moment that it is \(2.10 \mathrm{m}\) 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.

Step by step solution

01

Calculate the Time of Flight Until the Ball Reaches 2.10 meters

We start by finding the time it takes for the ball to reach its peak height. Using the vertical motion equation \( y = v_{0y}t + \frac{1}{2}at^2 \), where \( y = 2.10 \) m (the height the ball needs to reach), initial vertical speed \( v_{0y} = 15 \sin(50^{\circ}) \), and \( a = -9.81 \ m/s^2 \) (acceleration due to gravity). Solving for \( t \) gives us two values; one is the time when the ball is ascending, and another is when it is descending.

02

Find the Horizontal Distance the Ball Travels

We calculate the horizontal velocity component \( v_{0x} = 15 \cos(50^{\circ}) \ m/s \). Then, use the flight time obtained from Step 1 to calculate the range of the ball using \( x = v_{0x}t \).

03

Calculate the Time Delay and Effective Time for Opponent Movement

Since the opponent starts moving 0.30 seconds after the ball is launched, we subtract 0.30 seconds from the total flight time of the ball to get the effective time that the opponent has to reach the ball.

04

Calculate the Required Speed of the Opponent

The opponent must cover a distance of 10 meters in the reduced time calculated in Step 3. We use the equation \( \text{speed} = \frac{\text{distance}}{\text{time}} \) to find the minimum average speed the opponent must move at to reach the ball.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Tennis Physics

Understanding the physics of tennis can give players a strategic edge. In sports like tennis, physics principles help explain the motion of the ball and how it interacts with the players. One key aspect is projectile motion, which is essential for maneuvers like the lob.

• A lob involves launching the ball at an angle, creating an arc over an opponent's head.
• Successful lobs depend on the ball's initial velocity, launch angle, and the gravitational force.

Each element determines how high and how far the ball will travel. Mastering these details can significantly improve a player's game strategy.

Vertical Motion

Vertical motion examines how an object moves under the influence of gravity. For the tennis lob, this means analyzing how the ball rises and falls after being struck.

• The vertical motion is driven by the ball's initial vertical speed component and gravity.
• Using the formula: \[ y = v_{0y}t + \frac{1}{2}at^2 \] allows calculation of the ball's behavior over time.
• In this scenario, the ball needs to reach a height of 2.10 meters from its initial launch position.

Understanding these variables helps in predicting the peak height and duration the ball will be airborne.

Horizontal Motion

Horizontal motion is concerned with how far the ball travels before the opponent can return it. While gravity affects vertical motion, the initial horizontal velocity determines how far it will go.

• The horizontal component can be calculated using: \[ v_{0x} = 15\cos(50^\circ) \].
• This value stays constant as there's no horizontal acceleration in ideal conditions.
• Knowing the total time of flight helps in finding the range of the ball through \[ x = v_{0x}t \].

This helps in strategizing how far and fast an opponent needs to move.

Average Speed Calculation

Calculating average speed is crucial for determining how fast the opponent needs to move to reach the lobbed ball.

• Begin by noting the time delay of 0.30 seconds before the opponent starts moving.
• Calculate the opponent's effective movement time by subtracting the delay from the ball's total flight time.
• Use the formula: \[ \text{speed} = \frac{\text{distance}}{\text{time}} \] to find the required speed.
• This gives the minimum average speed necessary for the opponent to cover the 10 meters distance.

Knowing how to determine this speed enables players to anticipate actions in fast-paced games.