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 60

Chapter 2, Problem 60

For an object falling freely from rest, show that the distance traveled during each successive second increases in the ratio of successive odd integers (1, 3, 5, etc.). (This was first shown by Galileo.) See Figs. 2–27 and 2–30.

Verified step by step guidance

Start by recalling the kinematic equation for the distance traveled by an object under constant acceleration:

s = v_0t + (1/2)at^2

. Since the object is falling freely from rest, the initial velocity

v_0

is 0, and the equation simplifies to

s = (1/2)gt^2

, where

g

is the acceleration due to gravity.

To find the distance traveled during each successive second, calculate the distance traveled at the end of time

t

and subtract the distance traveled at the end of time

t-1

. This gives the distance traveled in the

t

-th second:

s_t = (1/2)g t^2 - (1/2)g (t-1)^2

Simplify the expression for

s_t

: Expand the squares to get

s_t = (1/2)g (t^2 - (t^2 - 2t + 1))

. Simplify further to obtain

s_t = (1/2)g (2t - 1)

, which represents the distance traveled during the

t

-th second.

Observe that the distance traveled during successive seconds is proportional to the sequence

2t - 1

, which generates the odd integers: 1, 3, 5, etc., as

t

increases.

Conclude that the distance traveled during each successive second increases in the ratio of successive odd integers (1, 3, 5, etc.), as shown by Galileo. This result arises from the quadratic dependence of distance on time in uniformly accelerated motion.

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Key Concepts

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

Free Fall

Free fall refers to the motion of an object under the influence of gravity alone, with no other forces acting on it, such as air resistance. In this scenario, the object accelerates downward at a constant rate, approximately 9.81 m/s² on Earth. Understanding free fall is crucial for analyzing the motion of objects in gravitational fields.

Distance Traveled in Free Fall

The distance traveled by an object in free fall can be calculated using the equation d = 1/2 g t², where d is the distance, g is the acceleration due to gravity, and t is the time in seconds. This equation shows that the distance increases with the square of the time, leading to greater distances covered in successive seconds.

Odd Integer Ratios

The concept of odd integer ratios arises from the distances covered in each second of free fall. Specifically, the distance traveled in the first second is 1 unit, in the second second is 3 units, and in the third second is 5 units, forming the sequence of odd integers. This pattern illustrates how the distance increases non-linearly, emphasizing the accelerating nature of free fall.

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Related Practice

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A baseball pitcher throws a baseball with a speed of 43 m/s. Estimate the average acceleration of the ball during the throwing motion. In throwing the baseball, the pitcher accelerates it through a displacement of about 3.5 m, from behind the body to the point where it is released (Fig. 2–44).

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