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Chapter 4: Problem 6

Is is possible to envision a motion where you for a period have no displacement, but non-zero velocity? (You may use an \(x\) ( \(t\) ) plot for illustration).

Short Answer

Expert verified
Yes, it is possible if the object moves and returns to the starting point within the period.

Step by step solution

01

- Understand Displacement and Velocity

Displacement is the overall change in position of an object. Velocity is the rate of change of displacement with respect to time. To have a non-zero velocity, the object must be moving, causing a change in displacement over some period.
02

- Analyze the Concept of Zero Displacement

Zero displacement over a period means that the object's starting position and ending position are the same for that period. This can happen if the object moves out and then returns to the starting point within that time frame.
03

- Explore Motion with Zero Displacement and Non-Zero Velocity

Consider an object moving in a back-and-forth manner such as a pendulum or an oscillating spring. The object continuously changes position (non-zero velocity) but can have zero net displacement over a period (it starts and ends at the same point).
04

- Visualize with an Example

Imagine a particle moving along the x-axis that starts at position 0, moves to position +5 units, then back to position 0 over a period. The displacement is zero because it starts and ends at the same point, but the velocity is non-zero as it is moving during the period.
05

- Illustrate with an x(t) Plot

Plot the position of the particle, x(t), over time, t. For the period where the particle moves from 0 to +5 and back to 0, the plot will show an increase from 0 to +5 followed by a decrease back to 0. The slope of the plot (velocity) is non-zero except at the turning points.

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Key Concepts

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

Displacement
Displacement is a key idea in understanding motion. It measures the overall change in position of an object. Imagine you move from your house to the grocery store and back. If your house is the start and end point, your displacement is zero, even though you might have walked many steps. Displacement considers only the initial and final positions, ignoring the path taken in between. It is a vector quantity, which means it has both magnitude and direction. For example, moving 5 meters east is different from moving 5 meters west.
Velocity
Velocity is the rate at which displacement changes with time. It tells us how fast and in what direction an object is moving. Unlike speed, which is scalar and only gives magnitude, velocity provides information about direction too. If a car moves 100 meters north in 10 seconds, its velocity is \(10 \, \text{m/s} \, \text{north}\). Velocity can change even if speed is constant. For example, if you move in a circle at a constant speed, your direction keeps changing, hence your velocity changes.
Motion Analysis
Motion analysis involves studying how objects move. We use graphs and equations to represent and predict motion. One common tool is the x(t) plot, which graphs position (x) over time (t). If the graph is a straight line, the velocity is constant. A curved line indicates changing velocity. Motion can be uniform or non-uniform. Uniform motion has constant velocity, while non-uniform motion has changing velocity. By analyzing these plots, we understand how position and velocity change over time.
Physics Concepts
Understanding displacement and velocity requires grasping some fundamental physics concepts.

  • Kinematics: Branch of physics that describes motion.
  • Vectors: Quantities that have both magnitude and direction, like displacement and velocity.
  • Scalars: Quantities with only magnitude, like speed and distance.

Combining these concepts allows us to analyze and predict the motion of objects.
Oscillatory Motion
Oscillatory motion refers to movement that repeats itself over time, like a pendulum or a spring. These systems move back and forth around an equilibrium position. Even though the object is constantly moving (non-zero velocity), it can have a zero displacement if it returns to its starting point. Imagine a pendulum swinging from left to right and back. Its displacement could be zero at specific times, even though it is in motion. This demonstrates how zero displacement doesn't always mean no movement.

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Most popular questions from this chapter

A car is driving along a straight road. Sketch the position and velocity as a function of time for the car if: (a) The car drives with constant velocity. (b) The car accelerates with a constant acceleration. (c) The car brakes with a constant acceleration.

A freight train travels from Oslo to Drammen at a velocity of \(50 \mathrm{~km} / \mathrm{h}\). An express train travels from Drammen to Oslo at \(200 \mathrm{~km} / \mathrm{h}\). Assume that the trains leave at the same time. The distance from Oslo to Drammen along the railway track is \(50 \mathrm{~km}\). You can assume the motion to a long a line. (a) When do the trains meet? (b) How far from Oslo do the trains meet?

An electron is shot through a box containing a constant electric field, getting accelerated in the process. The acceleration inside the box is \(a=2000 \mathrm{~m} / \mathrm{s}^{2}\). The width of the box is \(1 \mathrm{~m}\) and the electron enters the box with a velocity of \(100 \mathrm{~m} / \mathrm{s}\). (a) What is the velocity of the electron when it exits the box?

When you brake your car with your brand new tyres, your acceleration is \(5 \mathrm{~m} / \mathrm{s}^{2}\). (a) Find an expression for the distance you need to stop the car as a function of the starting velocity. With your old tires, the acceleration is only two thirds of the acceleration with the new tyres. (b) How does this affect the braking distance? (c) Your reaction time is \(0.5 \mathrm{~s}\). If a child jumps into the street \(30 \mathrm{~m}\) ahead of you when you are driving \(50 \mathrm{~km} / \mathrm{h}\), are you able to stop with your new tires? What would happen if you did not change tyres?

Your roommate sets off early to school, walking leisurely at \(0.5 \mathrm{~m} / \mathrm{s}\). Thirty minutes after she left, you realize that she forgot her lecture notes. You decide to run after her to give her the notes. You run at a healthy \(3 \mathrm{~m} / \mathrm{s}\). (a) What is her position when you start running? (b) What is your position when \(tt_{1}\). (k) How can you use this result to find where you catch up with your roommate? (1) Where do you catch up with your roommate? (m) What parts of your solution strategy are general, that is, what parts of your strategy do not change if we change how either person moves?
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