Ch 03: Motion in Two or Three Dimensions

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610당신이 사용하는 게 아니라요?교과서 변경

Young & Freedman Calc 14th Edition

Ch 03: Motion in Two or Three Dimensions

문제 10

3장, 문제 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?

검증된 단계별 안내

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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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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.

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.