Ch. 02 - Describing Motion: Kinematics in One Dimension

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

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Giancoli Douglas 5th edition

Ch. 02 - Describing Motion: Kinematics in One Dimension

Problem 87b

Chapter 2, Problem 87b

Two children are playing on two trampolines. The first child bounces up one-and-a-half times higher than the second child. The initial speed upwards of the second child is 4.0 m/s. What is the initial speed of the first child?

Verified step by step guidance

Step 1: Recall the kinematic equation for vertical motion under constant acceleration due to gravity:

h = \(\frac{v_0^2}{2g}\)

, where

h

is the maximum height,

v_0

is the initial velocity, and

g

is the acceleration due to gravity (approximately

9.8 \ \(\text{m/s}\)^2

Step 2: Let the maximum height reached by the second child be

h_2

. Using the given initial velocity of the second child,

v_{0,2} = 4.0 \ \(\text{m/s}\)

, substitute into the equation:

h_2 = \(\frac{(4.0)^2}{2 \cdot 9.8}\)

. This gives the height

h_2

in terms of known values.

Step 3: The problem states that the first child bounces one-and-a-half times higher than the second child. Therefore, the maximum height of the first child is

h_1 = 1.5 \(\cdot\) h_2

. Substitute

h_2

from Step 2 into this equation to express

h_1

in terms of known values.

Step 4: Using the same kinematic equation for the first child,

h_1 = \(\frac{v_{0,1}\)^2}{2g}

, solve for the initial velocity of the first child:

v_{0,1} = \(\sqrt{2g \cdot h_1}\)

. Substitute

h_1

from Step 3 into this equation.

Step 5: Simplify the expression for

v_{0,1}

to find the initial velocity of the first child in terms of the given values. This will provide the relationship between the initial velocities of the two children.

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Key Concepts

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

Kinematics

Kinematics is the branch of mechanics that deals with the motion of objects without considering the forces that cause the motion. It involves concepts such as displacement, velocity, and acceleration. In this scenario, understanding kinematics is essential to relate the initial speed of the second child to the height achieved by both children on the trampolines.

Potential Energy and Height

Potential energy is the energy stored in an object due to its position in a gravitational field, which is directly related to its height. The higher an object is raised, the more potential energy it possesses. In this case, the height reached by the first child, who bounces higher than the second, can be calculated using the relationship between initial speed and potential energy at the peak of the bounce.

Energy Conservation

The principle of energy conservation states that energy cannot be created or destroyed, only transformed from one form to another. In the context of the trampolines, the kinetic energy of the children at the moment of takeoff is converted into potential energy at the peak of their jumps. This principle allows us to relate the initial speeds of both children based on the heights they achieve.

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