Problem 18 - Thomas Calculus

Question

Can anything be said about the acceleration of a particle that is moving at a constant speed? Give reasons for your answer.

Step-by-Step Solution

Answer

The acceleration is zero for straight-line constant speed and non-zero (centripetal) for constant speed in a curve.

Step 1: Understanding Constant Speed

When a particle is moving at a constant speed, it means that the magnitude of its velocity remains unchanged over time. This tells us that the particle does not speed up or slow down.

Step 2: Defining Acceleration

Acceleration is defined as the rate at which an object's velocity changes over time. When we talk about acceleration, we look at the change in both the magnitude and direction of the velocity vector.

Step 3: Analyzing Constant Velocity

If a particle's speed is constant, and its path is straight, its velocity remains constant, and thus it experiences no change in velocity, meaning it has zero acceleration.

Step 4: Considering Directional Change

Even though the speed is constant, if the particle is moving along a curved path, the direction of velocity changes continuously. A change in the direction of velocity is also a form of acceleration, called centripetal acceleration.

Step 5: Conclusion

If the particle moves in a straight line at a constant speed, its acceleration is zero. However, if it moves in a curved path at a constant speed, the particle has non-zero (centripetal) acceleration due to changes in direction.

Key Concepts

Acceleration

Acceleration tells us how quickly a particle's velocity is changing. It looks at both the size (called magnitude) and the direction of that change. When we think about something speeding up or slowing down, it's the magnitude of velocity that changes. But remember, acceleration can also occur if the direction of movement changes.

To sum up,

• If only the magnitude of velocity changes, this is linear acceleration.
• If only the direction of movement changes, this is typically referred to as centripetal acceleration—more on that later.

It's important to keep in mind that acceleration is about any change in velocity, whether that's speeding up, slowing down, or simply changing direction.

Velocity

Velocity is more than just speed; it includes direction too. Think of velocity as a vector, representing both how fast something is going and where it's headed. For example, a car driving north at 60 km/h has a different velocity than a car driving south at the same speed.

In physics, when we analyze velocity:

• "Magnitude" represents how fast the object is moving.
• "Direction" shows the path the object follows.

If an object moves at a constant velocity, it means both speed and direction remain unchanged. However, variations in either will lead to changes in velocity, which involves acceleration.

Centripetal Acceleration

Centripetal acceleration occurs when an object moves in a curved path. Even if the speed remains constant, because the direction changes, the object is undergoing acceleration. This special type of acceleration is called centripetal, which means "center-seeking."

The key points about centripetal acceleration are:

• It always points toward the center of the circle that the object is moving along.
• It doesn't change the speed of the object, only its direction.

For instance, when you swing a ball tied to a string, the ball experiences centripetal acceleration towards your hand, keeping it moving along a circular path.

Constant Velocity

Constant velocity means that both the speed and direction of an object remain unchanged. It is an indication that there is no net force acting on the object. Thus, in the absence of external forces, it will continue its motion in a straight line.

To understand constant velocity better, consider:

• An object moving at constant velocity doesn't accelerate, so its acceleration is zero.
• If there is a change in direction or speed, it will no longer be at constant velocity.

This means maintaining a constant velocity perfectly aligns with an unchanging state of motion, unless an external force intervenes.