Problem 19 - Thomas Calculus

Question

Can anything be said about the speed of a particle whose acceleration is always orthogonal to its velocity? Give reasons for your answer.

Step-by-Step Solution

Answer

The speed of the particle remains constant, as acceleration only changes velocity direction, not magnitude.

Step 1: Understanding the Problem

We are given a particle whose acceleration is always orthogonal (perpendicular) to its velocity. We need to deduce something about the speed of the particle from this information.

Step 2: Analyze Orthogonal Relationship

If acceleration is orthogonal to velocity, it means that the dot product of the acceleration vector \( \mathbf{a} \) and the velocity vector \( \mathbf{v} \) is zero: \( \mathbf{a} \cdot \mathbf{v} = 0 \).

Step 3: Understanding Speed

Speed is the magnitude of the velocity vector \( \mathbf{v} \). Thus, speed is computed as \( ||\mathbf{v}|| = \sqrt{v_x^2 + v_y^2 + v_z^2} \) if the velocity has components \( v_x, v_y, v_z \).

Step 4: Relationship Between Acceleration and Speed

Since \( \mathbf{a} \cdot \mathbf{v} = 0 \), acceleration only changes the direction of the velocity vector, not its magnitude. Hence, any change in velocity due to acceleration will maintain the speed constant.

Step 5: Conclusion

Therefore, if a particle's acceleration is always orthogonal to its velocity, the speed of the particle remains constant over time.

Key Concepts

Speed Constancy

When we talk about speed constancy, we're referring to how the speed of an object stays the same over time. This is different from velocity, which includes both speed and direction. For a particle moving in a path where its acceleration is orthogonal to its velocity, the speed remains constant. Why is that? Because, when acceleration is perpendicular to velocity, it doesn't add or subtract any speed. Instead, it only changes direction.

• Imagine running at a steady speed but only turning left or right—your speed doesn't change, just your path.
• This is the essence of speed constancy in this context.

Understanding this helps you grasp why certain objects maintain the same speed until other forces affect them.

Dot Product and Orthogonality

The dot product is a mathematical tool that tells us about the angle between two vectors. Two vectors are orthogonal if their dot product is zero. That means they are at a 90-degree angle to each other, like the axes on a graph.
For the case of velocity and acceleration:

• The velocity vector tells us how fast something is moving and in which direction.
• The acceleration vector tells us how the velocity is changing.
• If they are orthogonal, or perpendicular, their dot product gives zero: \( \mathbf{a} \cdot \mathbf{v} = 0 \).

This relationship holds the key to understanding why speed remains constant; without a component of acceleration in the direction of velocity, the speed does not change.

Vector Magnitude

A vector magnitude is essentially the length, or size, of the vector. Think of it like the length of an arrow on a map, illustrating both how far you'll go and in what direction.
For velocity, the magnitude of the vector is the speed of the particle. It can be calculated using the formula:
\[||\mathbf{v}|| = \sqrt{v_x^2 + v_y^2 + v_z^2} \]where \( v_x, v_y, \text{and} \ v_z \) are the vector components.

• The magnitude remains constant if the vector changes direction but not length.
• This is exactly what happens when acceleration and velocity are orthogonal.

Understanding vector magnitude is crucial for linking physics concepts to mathematics.

Velocity and Acceleration Relationship

The relationship between velocity and acceleration describes how an object's movement changes over time. Normally, acceleration can change both the speed and direction of an object. However, in scenarios where acceleration is always orthogonal to velocity, the changes only affect direction.

• This means: the "forward push" or "backward drag" is zero, causing no speed change.
• But, the object could be spinning or curving without speeding up or slowing down.

In essence, if acceleration doesn't have any component along the velocity vector, the speed is not affected, explaining why the speed remains constant and only direction changes.