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Ch 03: Motion in Two or Three Dimensions
Young & Freedman Calc - University Physics 14th Edition
Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoYoung & Freedman Calc 14th EditionCh 03: Motion in Two or Three DimensionsProblema 14a
Capitolo 3, Problema 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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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.
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Introduction 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.
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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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