Ch 03: Motion in Two or Three Dimensions

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook

Young & Freedman Calc 14th Edition

Ch 03: Motion in Two or Three Dimensions

Problem 10

Chapter 3, Problem 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?

Verified step by step guidance

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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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 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.

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Vertical Motion & Free Fall

Related Practice

Textbook Question

A remote-controlled car is moving in a vacant parking lot. The velocity of the car as a function of time is given by $\vec{v}\) = $\left$[ 5.00~$\mathrm{m/s}$ - (0.0180~$\mathrm{m/s^3}$)t^2 $\right$] $\hat{i}$ + $\left$[ 2.00~$\mathrm{m/s}$ + (0.550~$\mathrm{m/s^2}$)t $\right$] \(\hat{j}$ . What are the magnitude and direction of the car's velocity at $t equals 8.00 s$ ? (b) What are the magnitude and direction of the car's acceleration at $t equals 8.00 s$ ?

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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?

A remote-controlled car is moving in a vacant parking lot. The velocity of the car as a function of time is given by $\vec{v}\) = $\left$[ 5.00~$\mathrm{m/s}$ - (0.0180~$\mathrm{m/s^3}$)t^2 $\right$] $\hat{i}$ + $\left$[ 2.00~$\mathrm{m/s}$ + (0.550~$\mathrm{m/s^2}$)t $\right$] \(\hat{j}$ . What are $a_{x} (t)$ and $a_{y} (t)$ , the $x$ - and $y$ - components of the car's velocity as functions of time?

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Textbook Question

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Textbook Question

A physics book slides off a horizontal tabletop with a speed of 1.10 m/s. It strikes the floor in 0.480 s. Ignore air resistance. Find the height of the tabletop above the floor.

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