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

Young & Freedman Calc 14th Edition

Ch 03: Motion in Two or Three Dimensions

Problema 10

Capitolo 3, Problema 10

A daring 510 N swimmer dives off a cliff with a running horizontal leap, as shown in Fig. E3.10. What must her minimum speed be just as she leaves the top of the cliff so that she will miss the ledge at the bottom, which is 1.75 m wide and 9.00 m below the top of the cliff?

Guida verificata passo dopo passo

Identify the problem as a projectile motion problem where the swimmer dives horizontally off a cliff. The goal is to find the minimum horizontal speed required to clear the ledge.

Use the kinematic equation for vertical motion to determine the time it takes for the swimmer to fall 9.00 m. The equation is:

y = \frac{1}{2} g t^{2}

, where

y

is the vertical distance,

g

is the acceleration due to gravity (9.81 m/s²), and

t

is the time.

Solve the equation for

t

to find the time it takes to fall 9.00 m. Rearrange the equation to:

t = \sqrt{\frac{2 y}{g}}

Use the time calculated to determine the minimum horizontal speed required to clear the ledge. The horizontal distance covered is given by:

x = v

₀t, where

x

is the horizontal distance (1.75 m) and

v

₀ is the initial horizontal speed.

Rearrange the equation to solve for

v

₀:

v

₀=xt. Substitute the values for

x

and

t

to find the minimum speed.

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Projectile Motion

Projectile motion refers to the motion of an object that is launched into the air and is subject to the force of gravity. In this scenario, the swimmer's horizontal leap can be analyzed as a projectile, where the horizontal and vertical motions are independent. The horizontal distance covered and the time taken to fall can be calculated to determine the swimmer's minimum speed.

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Percorso guidato

04:44

Introduction to Projectile Motion

Kinematics Equations

Kinematics equations describe the motion of objects under constant acceleration, such as gravity. For the swimmer, the vertical motion can be analyzed using the equation for free fall, which relates distance, initial velocity, acceleration, and time. This allows us to calculate the time it takes to fall 9.00 m, which is crucial for determining the horizontal speed needed to clear the ledge.

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Percorso guidato

08:25

Kinematics Equations

Horizontal and Vertical Components of Motion

In projectile motion, the horizontal and vertical components of motion are analyzed separately. The horizontal component remains constant (assuming no air resistance), while the vertical component is influenced by gravity. Understanding how to separate these components is essential for calculating the swimmer's minimum horizontal speed to ensure she clears the 1.75 m ledge while falling 9.00 m.

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Percorso guidato

08:36

Vertical Motion & Free Fall

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