Ch 02: Motion Along a Straight Line

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 02: Motion Along a Straight Line

Problem 36b

Chapter 2, Problem 36b

A small rock is thrown vertically upward with a speed of $22.0$ m/s from the edge of the roof of a $30.0$ -m-tall building. The rock doesn't hit the building on its way back down and lands on the street below. Ignore air resistance. How much time elapses from when the rock is thrown until it hits the street?

Verified step by step guidance

Identify the initial conditions: The initial velocity of the rock is 22.0 m/s upward, and it is thrown from a height of 30.0 m above the ground.

Use the kinematic equation for vertical motion:

y = y_0 + v_0 t - \frac{1}{2} g t^2

, where

y

is the final position (0 m, since it lands on the street),

y_0

is the initial position (30.0 m),

v_0

is the initial velocity (22.0 m/s),

g

is the acceleration due to gravity (9.8 m/s²), and

t

is the time elapsed.

Set up the equation with the known values:

0 = 30.0 + 22.0 t - \frac{1}{2} 9.8 t^2

. This is a quadratic equation in terms of

t

Rearrange the equation to standard quadratic form:

\frac{1}{2} 9.8 t^2 - 22.0 t - 30.0 = 0

. This can be simplified to

4.9 t^2 - 22.0 t - 30.0 = 0

Solve the quadratic equation using the quadratic formula:

t = \frac{- b \pm \sqrt{b^2 - 4 a c}}{2 a}

, where

a

is 4.9,

b

is -22.0, and

c

is -30.0. Calculate the two possible values for

t

and choose the positive one, as time cannot be negative.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Kinematics Equations

Kinematics equations describe the motion of objects without considering the forces that cause the motion. They are essential for solving problems involving velocity, acceleration, and displacement. In this scenario, the equations help calculate the time taken for the rock to reach its peak height and descend to the street.

Free Fall Motion

Free fall motion refers to the movement of an object under the influence of gravity alone, with no other forces acting on it. The rock's upward and downward journey can be analyzed using the principles of free fall, where the acceleration due to gravity is approximately 9.81 m/s² downward.

Initial Velocity and Displacement

Initial velocity is the speed at which the rock is thrown upward, and displacement is the change in position from the starting point. Understanding these concepts is crucial for determining the total time of flight, as they influence the rock's trajectory and the time it takes to hit the street.

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Textbook Question

You throw a glob of putty straight up toward the ceiling, which is $3.60$ m above the point where the putty leaves your hand. The initial speed of the putty as it leaves your hand is $9.50$ m/s. How much time from when it leaves your hand does it take the putty to reach the ceiling?

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Textbook Question

A small rock is thrown vertically upward with a speed of $22.0$ m/s from the edge of the roof of a $30.0$ -m-tall building. The rock doesn't hit the building on its way back down and lands on the street below. Ignore air resistance. What is the speed of the rock just before it hits the street?

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