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 remains constant; only the direction of motion changes.

Step 1: Understanding Orthogonality

When acceleration is always orthogonal (perpendicular) to velocity, it means that at any given point in time, the vector of acceleration forms a 90-degree angle with the vector of velocity. This can be expressed mathematically as the dot product of the velocity vector \( \mathbf{v} \) and the acceleration vector \( \mathbf{a} \) being equal to zero, i.e., \( \mathbf{v} \cdot \mathbf{a} = 0 \).

Step 2: Implication of Orthogonal Vectors

If the dot product of two vectors is zero, it indicates that the vectors are orthogonal. For the vectors of velocity and acceleration, this orthogonality means that acceleration does not contribute to any increase or decrease in the magnitude of velocity. Instead, it only changes the direction.

Step 3: Effect on Speed

Speed is the magnitude of the velocity vector. Since acceleration is orthogonal to velocity, the magnitude or speed remains constant; only the direction of the velocity changes. Consequently, a perpendicular acceleration changes only the path of the particle, not its speed.

Step 4: Conclusion about Speed

If a particle's acceleration is always orthogonal to its velocity, the speed of the particle remains unchanged. Physically, this is akin to circular motion where the speed is constant and only the direction changes.

Key Concepts

Orthogonality

Orthogonality is a fundamental concept in vector mathematics that refers to the idea of vectors being perpendicular to one another. In the context of motion, if a particle's acceleration vector is orthogonal to its velocity vector, it means they form a 90-degree angle. This situation is represented mathematically using the dot product, which will equal zero. Orthogonality implies that the acceleration does not influence the magnitude of the velocity. Instead, it solely affects the direction of motion. This is crucial in scenarios such as circular motion, where the acceleration (centripetal acceleration) is always perpendicular to the velocity, ensuring the path's curvature.

Dot Product

The dot product is a simple yet powerful operation that tells us about the relationship between two vectors. It is calculated by multiplying the corresponding components of the vectors and summing the result. Mathematically, for two vectors \( \mathbf{a} \) and \( \mathbf{b} \), this is expressed as:

• \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 \)

If the dot product is zero, it indicates that these vectors are orthogonal, meaning they meet at right angles. In dynamics, when acceleration is orthogonal to velocity, the dot product becomes zero (\( \mathbf{v} \cdot \mathbf{a} = 0 \)). This zero result confirms that the acceleration does not change the speed, only influencing the direction in which the particle is moving.

Constant Speed

A constant speed denotes that the magnitude of the velocity vector of a particle remains unchanged over time. In physical terms, let's consider a scenario where acceleration is orthogonal to velocity. This orthogonality implies that though the particle experiences acceleration, it does not speed up or slow down. The path of the particle might change, but the speed stays the same. A real-world example of this is circular motion, where the speed is constant, but the path—a circle or curve—continuously changes due to the change in direction of the velocity.

Acceleration

Acceleration typically involves a change in the velocity of an object over time. However, when acceleration is orthogonal to velocity, it is unique in that it does not alter the speed (magnitude of velocity). This kind of acceleration alters only the direction, not the speed. Such acceleration is observable in circular motion, like that of a car taking a round turn. The car's speed isn't increasing or decreasing, yet it's experiencing a sideways force that changes the path it follows. Understanding this concept helps clarify why objects in circular paths can have constant speeds despite continuous acceleration.