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Chapter 2: Problem 1

Is it possible for speed to be constant while acceleration is not zero? Give an example of such a situation.

Short Answer

Expert verified
Yes, it's possible. An object moving in a circle at constant speed has a non-zero centripetal acceleration, because the direction of its velocity changes.

Step by step solution

01

Understand Speed and Acceleration

Speed refers to how fast an object is moving and is measured as the distance traveled per unit of time. Acceleration, on the other hand, measures the rate of change of speed per unit of time. For acceleration to be non-zero, the speed of an object must change either in magnitude or direction.
02

Identify Constant Speed with Changing Direction

A situation where an object can have constant speed while having a non-zero acceleration is circular motion. Here, while the object moves at a constant speed, the direction of the velocity changes continuously, leading to a centripetal acceleration.
03

Provide Example of Non-Zero Acceleration with Constant Speed

An example is a car moving with a constant speed around a circular track. While the speed (magnitude) remains constant, the direction of the velocity changes. Thus, there is a centripetal acceleration towards the center of the circle, which means acceleration is not zero.

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Key Concepts

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

Constant Speed
When we speak of an object moving at a constant speed, we mean that the object covers equal distances in equal intervals of time, regardless of the direction in which it is moving. In everyday terms, an object with constant speed is not getting faster or slower.

It's easy to mistake constant speed for a lack of acceleration, but they are not synonymous. In uniform circular motion, for instance, the speed of an object is constant because it travels the same distance around the circle in each second. However, because the object's direction is continually changing, it is accelerating despite the speedometer not showing any changes in speed.
Non-Zero Acceleration
The concept of non-zero acceleration can be a bit tricky to understand initially. Acceleration is a vector, meaning it has both a magnitude and a direction. It represents the rate at which an object's velocity changes. If a car is speeding up or slowing down, it's clear to us that the car is accelerating. However, even if the car maintains a steady speedometer reading but takes a turn, it is still accelerating due to the change in direction.

Thus, non-zero acceleration happens not just when the speed of an object changes, but also when the direction of motion changes. In the context of physics and particularly circular motion, it's crucial to note that a constant speed doesn't negate the possibility of acceleration.
Centripetal Acceleration
When an object is in circular motion, it experiences a special type of acceleration known as centripetal acceleration. This acceleration is always directed towards the center of the circular path. The formula to compute centripetal acceleration (\( a_c \) is \[ a_c = \frac{v^2}{r} \] where \( v \) is the speed of the object moving in a path of radius \( r \).

Despite the speed of the object remaining constant, the continuous change in direction requires a force to keep the object moving in that circle. This is where centripetal force comes in, causing the centripetal acceleration. It is the constant change in the velocity's direction that leads to this acceleration, not a change in speed.
Velocity Change Direction
The concept of velocity changing direction is an integral aspect of circular motion and indeed any curved path. While speed tells us how fast something is going, velocity specifies both the speed and direction of an object's movement.

When an object moves in a circle, even at a constant speed, its velocity is not constant because its direction changes at every instant. This is foundational to understanding why objects moving in circles have acceleration. The presence of an acceleration in scenarios where an object is changing direction but not speed is why we can have non-zero acceleration with a constant speed.

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Most popular questions from this chapter

Under what circumstances does distance traveled equal magnitude of displacement? What is the only case in which magnitude of displacement and displacement are exactly the same?

While entering a freeway, a car accelerates from rest at a rate of \(2.40 \mathrm{~m} / \mathrm{s}^{2}\) for \(12.0 \mathrm{~s}\). (a) Draw a sketch of the situation. (b) List the knowns in this problem. (c) How far does the car travel in those \(12.0\) s? To solve this part, first identify the unknown, and then discuss how you chose the appropriate equation to solve for it. After choosing the equation, show your steps in solving for the unknown, check your units, and discuss whether the answer is reasonable. (d) What is the car's final velocity? Solve for this unknown in the same manner as in part (c), showing all steps explicitly.

A student drove to the university from her home and noted that the odometer reading of her car increased by \(12.0 \mathrm{~km}\). The trip took \(18.0\) min. (a) What was her average speed? (b) If the straight-line distance from her home to the university is \(10.3 \mathrm{~km}\) in a direction \(25.0^{\circ}\) south of east, what was her average velocity? (c) If she returned home by the same path 7 h 30 min after she left, what were her average speed and velocity for the entire trip?

A cheetah can accelerate from rest to a speed of 30.0 m/s in 7.00 s. What is its acceleration?

Give an example in which there are clear distinctions among distance traveled, displacement, and magnitude of displacement. Specifically identify each quantity in your example.
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