Ch. 03 - Kinematics in Two or Three Dimensions; Vectors

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 03 - Kinematics in Two or Three Dimensions; Vectors

Problem 56

Chapter 3, Problem 56

At serve, a tennis player aims to hit the ball horizontally. What minimum speed is required for the ball to clear the 0.90-m-high net about 15.0 m from the server if the ball is 'launched' from a height of 2.30 m? Where will the ball land if it just clears the net (and will it be 'good' in the sense that it lands within 7.0 m of the net)? How long will it be in the air? See Fig. 3–50.

Verified step by step guidance

Step 1: Break the problem into horizontal and vertical motion components. Use the equations of motion for each direction separately. The horizontal motion is uniform (constant velocity), while the vertical motion is influenced by gravity (acceleration = -9.8 m/s²).

Step 2: For vertical motion, calculate the time it takes for the ball to drop from the initial height of 2.30 m to the height of the net (0.90 m). Use the kinematic equation:

y^{2} = y^{1} + v

_y0t + 12gt2, where

v

_y0 is 0 (horizontal launch). Solve for

t

Step 3: Use the time calculated in Step 2 to determine the horizontal velocity required for the ball to travel 15.0 m horizontally while clearing the net. Use the equation for horizontal motion:

x = v

_xt, where

v

_x is the horizontal velocity.

Step 4: To find where the ball lands, calculate the total time of flight. Use the vertical motion equation to find the time it takes for the ball to fall from 2.30 m to the ground (y = 0). Then, use this total time in the horizontal motion equation to find the total horizontal distance traveled.

Step 5: Compare the landing position of the ball to the net's position (15.0 m) and the court's boundary (7.0 m beyond the net). Determine if the ball lands within the 'good' range. Additionally, verify the total time the ball is in the air using the calculated values.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Projectile Motion

Projectile motion describes the motion of an object that is launched into the air and is subject to gravitational forces. It can be analyzed in two dimensions: horizontal and vertical. The horizontal motion is uniform, while the vertical motion is influenced by gravity, leading to a parabolic trajectory. Understanding this concept is crucial for determining the ball's path and the time it spends in the air.

Kinematic Equations

Kinematic equations relate the displacement, initial velocity, final velocity, acceleration, and time of an object in motion. For projectile motion, these equations can be used to calculate the time of flight, the maximum height, and the range of the projectile. In this scenario, they will help determine the minimum speed required for the ball to clear the net and where it will land.

Energy Conservation

The principle of energy conservation states that energy cannot be created or destroyed, only transformed from one form to another. In the context of projectile motion, the kinetic energy of the ball at launch converts to potential energy as it rises. Understanding this concept helps in analyzing the ball's speed and height at various points in its trajectory, which is essential for solving the problem of clearing the net.

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Related Practice

Textbook Question

Two cars approach a street corner at right angles to each other (Fig. 3–57). Car 1 travels at a speed relative to Earth v₁ₑ = 35 km/h, and car 2 at v₂ₑ = 55 km/h. What is the relative velocity of car 1 as seen by car 2? What is the velocity of car 2 relative to car 1?

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Textbook Question

A stunt driver wants to make his car jump over 8 cars parked side by side below a horizontal ramp (Fig. 3–46). With what minimum speed must he drive off the horizontal ramp? The vertical height of the ramp is 1.5 m above the car roofs and the horizontal distance he must clear is 22 m.

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Textbook Question

A person in the passenger basket of a hot-air balloon throws a ball horizontally outward from the basket with speed 12.0 m/s (Fig. 3–64). What initial velocity (magnitude and direction) does the ball have relative to a person standing on the ground if the hot-air balloon is descending at 3.0 m/s relative to the ground?

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Textbook Question

A motorboat whose speed in still water is 4.30 m/s must aim upstream at an angle of 23.5° (with respect to a line perpendicular to the shore) in order to travel directly across the stream. What is the resultant speed of the boat with respect to the shore? (See Fig. 3–33.)

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Textbook Question

A skier is accelerating down a 30.0° hill at 1.80 m/s² (Fig. 3–42). How long will it take her to reach the bottom of the hill, assuming she starts from rest and accelerates uniformly, if the elevation change is 125 m?

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Textbook Question

A diver running 2.5 m/s dives out horizontally from the edge of a vertical cliff and 3.5 s later reaches the water below. How high was the cliff and how far from its base did the diver hit the water?

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