Ch 03: Motion in Two or Three Dimensions

Young & Freedman Calc - University Physics 15th Edition

Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Not the one you use?Change textbook

Young & Freedman Calc 15th Edition

Ch 03: Motion in Two or Three Dimensions

Problem 14a

Chapter 3, Problem 14a

The froghopper, Philaenus spumarius, holds the world record for insect jumps. When leaping at an angle of 58.0° above the horizontal, some of the tiny critters have reached a maximum height of 58.7 cm above the level ground. (See Nature, Vol. 424, July 31, 2003, p. 509.) What was the takeoff speed for such a leap?

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Start by identifying the known values: the angle of the leap \( \theta = 58.0^{\circ} \) and the maximum height \( h = 58.7 \text{ cm} \). Convert the height to meters for consistency in units: \( h = 0.587 \text{ m} \).

Use the kinematic equation for vertical motion to relate the maximum height to the initial vertical velocity component \( v_{0y} \). The equation is \( v_{0y}^2 = 2gh \), where \( g \) is the acceleration due to gravity \( 9.81 \text{ m/s}^2 \). Solve for \( v_{0y} \).

Calculate the initial vertical velocity component \( v_{0y} \) using the equation \( v_{0y} = \sqrt{2gh} \). Substitute the values for \( g \) and \( h \) to find \( v_{0y} \).

Relate the initial vertical velocity component \( v_{0y} \) to the takeoff speed \( v_0 \) using the angle of the leap. The relationship is \( v_{0y} = v_0 \sin(\theta) \). Rearrange to solve for \( v_0 \): \( v_0 = \frac{v_{0y}}{\sin(\theta)} \).

Substitute the calculated \( v_{0y} \) and the angle \( \theta \) into the equation \( v_0 = \frac{v_{0y}}{\sin(\theta)} \) to find the takeoff speed \( v_0 \). Ensure the angle is in radians if using a calculator.

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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 refers to the motion of an object thrown or projected into the air, subject to only the acceleration of gravity. It involves two components: horizontal and vertical motion, which are independent of each other. Understanding the initial velocity, angle of projection, and maximum height is crucial for solving problems related to projectile motion.

Kinematic Equations

Kinematic equations describe the motion of objects in terms of displacement, velocity, acceleration, and time. For projectile motion, these equations help determine the relationship between the initial velocity, angle of projection, and maximum height. The equation for vertical motion, h = (v^2 * sin^2(θ)) / (2g), is particularly useful for finding the takeoff speed when the maximum height is known.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are essential for resolving the initial velocity into horizontal and vertical components in projectile motion. The sine function is used to calculate the vertical component (v * sin(θ)), while the cosine function is used for the horizontal component (v * cos(θ)). These components are crucial for analyzing the motion and solving for unknown variables like takeoff speed.

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Related Practice

Textbook Question

A rookie quarterback throws a football with an initial upward velocity component of 12.0 m/s and a horizontal velocity component of 20.0 m/s. Ignore air resistance. How much time (after it is thrown) is required for the football to return to its original level? How does this compare with the time calculated in part (a)?

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Textbook Question

On level ground a shell is fired with an initial velocity of 40.0 m/s at 60.0° above the horizontal and feels no appreciable air resistance. How long does it take the shell to reach its highest point?

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Textbook Question

A rookie quarterback throws a football with an initial upward velocity component of 12.0 m/s and a horizontal velocity component of 20.0 m/s. Ignore air resistance. How high is this point?

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Textbook Question

A rookie quarterback throws a football with an initial upward velocity component of 12.0 m/s and a horizontal velocity component of 20.0 m/s. Ignore air resistance. How far has the football traveled horizontally during this time?

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Textbook Question

On level ground a shell is fired with an initial velocity of 40.0 m/s at 60.0° above the horizontal and feels no appreciable air resistance. Find the horizontal and vertical components of the shell's initial velocity.

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Textbook Question

On level ground a shell is fired with an initial velocity of 40.0 m/s at 60.0° above the horizontal and feels no appreciable air resistance. Find its maximum height above the ground.

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