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Chapter 2: Problem 4

(I) A rolling ball moves from \(x_1 =\) 8.4 cm to \(x_2 = -\)4.2 cm during the time from \(t_1 =\) 3.0 s to \(t_2 =\) 6.1 s. What is its average velocity over this time interval?

Short Answer

Expert verified
The average velocity of the ball is approximately -4.06 cm/s.

Step by step solution

01

Understand the Formula for Average Velocity

The formula for average velocity is given by the change in position (displacement) divided by the change in time. It can be written as: \( v_{avg} = \frac{\Delta x}{\Delta t} \) where \( \Delta x \) is the change in position and \( \Delta t \) is the change in time.
02

Calculate the Change in Position (Displacement)

Find the displacement \( \Delta x \) by subtracting the initial position from the final position: \( \Delta x = x_2 - x_1 = -4.2 \text{ cm} - 8.4 \text{ cm} \). Simplifying, we get \( \Delta x = -12.6 \text{ cm} \).
03

Calculate the Change in Time

Find the change in time \( \Delta t \) by subtracting the initial time from the final time: \( \Delta t = t_2 - t_1 = 6.1 \text{ s} - 3.0 \text{ s} \). Simplifying, we get \( \Delta t = 3.1 \text{ s} \).
04

Compute the Average Velocity

Use the average velocity formula with the values found in previous steps: \( v_{avg} = \frac{-12.6 \text{ cm}}{3.1 \text{ s}} \). Calculate to find \( v_{avg} \approx -4.06 \text{ cm/s} \).
05

Interpret the Result

The negative sign in the average velocity indicates that the ball is moving in the direction from a higher to a lower position (from 8.4 cm to -4.2 cm) over the time interval.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Displacement
Displacement is an important concept when it comes to understanding how far an object has moved from its starting position. It is not just about the distance traveled, but rather the change in position from the start point to the endpoint.

In the context of our exercise, displacement is calculated by taking the final position of the ball (\(x_2 = -4.2\text{ cm}\)) and subtracting the initial position (\(x_1 = 8.4\text{ cm}\)). This is expressed as:
  • \( \Delta x = x_2 - x_1 \)
  • \( \Delta x = -4.2 - 8.4 \)
  • \( \Delta x = -12.6 \text{ cm} \)
The resulting displacement is \(-12.6\text{ cm}\), which is a negative value. This indicates that the ball moved backwards, or in the opposite direction of the positive coordinate axis.
Time Interval
The time interval is the duration over which an object's movement is measured. It's important to note that the time interval is a scalar quantity, meaning it doesn't have a direction like displacement does.

To find the time interval \(\Delta t\) for the ball in our exercise, we subtract the initial time from the final time. This calculation looks like:
  • \( \Delta t = t_2 - t_1 \)
  • \( \Delta t = 6.1 \text{ s} - 3.0 \text{ s} \)
  • \( \Delta t = 3.1 \text{ s} \)
The time interval is \(3.1\text{ s}\), showing how long it took for the ball to move from its initial to its final position.
Velocity Calculation
Velocity calculation is key to understanding an object's motion. Average velocity gives us the total displacement divided by the total time. It indicates how fast an object changes its position over a period of time and in which direction.

The formula used for calculating average velocity is:
  • \( v_{avg} = \frac{\Delta x}{\Delta t} \)
For our ball exercise, this would be:
  • \( v_{avg} = \frac{-12.6 \text{ cm}}{3.1 \text{ s}} \)
  • \( v_{avg} \approx -4.06 \text{ cm/s} \)
The negative sign in the calculated average velocity \(-4.06 \text{ cm/s}\) reveals that the ball is moving in the opposite direction to the positive direction. This is an essential aspect because it not only tells us how fast the ball is moving across a specific distance, but also that it's moving backward from where it started.

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Most popular questions from this chapter

(II) Digital bits on a 12.0-cm diameter audio CD are encoded along an outward spiraling path that starts at radius \(R_1 =\) 2.5 cm and finishes at radius \(R_2 =\) 5.8 cm. The distance between the centers of neighboring spiralwindings is 1.6 \(\mu\)m (= 1.6 \(\times\) 10\(^{-6}\) m). (\(a\)) Determine the total length of the spiraling path. [\(Hint\): Imagine "unwinding" the spiral into a straight path of width 1.6\(\mu\)m and note that the original spiral and the straight path both occupy the same area.] (\(b\)) To read information, a CD player adjusts the rotation of the CD so that the player's readout laser moves along the spiral path at a constant speed of about 1.2 m/s. Estimate the maximum playing time of such a CD.

(III) A fugitive tries to hop on a freight train traveling at a constant speed of 5.0 m/s. Just as an empty box car passes him, the fugitive starts from rest and accelerates at \(a =\) 1.4 m/s\(^2\) to his maximum speed of 6.0 m/s, which he then maintains. (\(a\)) How long does it take him to catch up to the empty box car? (\(b\)) What is the distance traveled to reach the box car?

(I) A bird can fly 25 km/h. How long does it take to fly 3.5 km?

Suppose a car manufacturer tested its cars for front-end collisions by hauling them up on a crane and dropping them from a certain height. (\(a\)) Show that the speed just before a car hits the ground, after falling from rest a vertical distance \(H\), is given by \(\sqrt{ 2gH }\) . What height corresponds to a collision at (\(b\)) 35 km/h? (\(c\)) 95 km/h?

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 up of the second child is 4.0 m/s. (\(a\)) Find the maximum height the second child reaches. (\(b\)) What is the initial speed of the first child? (\(c\)) How long was the first child in the air?
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