Question

Show that if the speed of a moving particle is constant its acceleration vector is always perpendicular to its velocity vector.

Short answer

The acceleration vector is always perpendicular to the velocity vector when speed is constant.

Step-by-step solution

Understand the problem statement

We need to demonstrate that if a particle's speed is constant, its acceleration is perpendicular to its velocity. This involves showing the relationship between speed, velocity, and acceleration vectors.

Define the velocity and acceleration vectors

Let \( \mathbf{v}(t) \) be the velocity vector of the particle as a function of time, and \( \mathbf{a}(t) \) be the acceleration vector. Speed is the magnitude of the velocity vector, noted as \( |mathbf{v}(t)| \).

Express constant speed in terms of velocity

The speed being constant means that \( |mathbf{v}(t)| = c \) for some constant \( c \). This implies that the derivative of the speed must be zero, \( \frac{d}{dt} |mathbf{v}(t)| = 0 \).

Differentiate the magnitude of the velocity vector

To differentiate \( |mathbf{v}(t)| \), use the relation \( |mathbf{v}(t)| = \sqrt{mathbf{v}(t) \cdot mathbf{v}(t)} \). Taking the derivative gives: \[ \frac{d}{dt} |mathbf{v}(t)| = \frac{1}{2\sqrt{mathbf{v}(t) \cdot mathbf{v}(t)}} 2 mathbf{v}(t) \cdot rac{d}{dt} mathbf{v}(t) = \frac{mathbf{v}(t) \cdot mathbf{a}(t)}{|mathbf{v}(t)|} = 0. \]

Conclude orthogonality from zero dot product

Since the dot product \( mathbf{v}(t) \cdot mathbf{a}(t) = 0 \), it indicates that vectors \( mathbf{v}(t) \) and \( mathbf{a}(t) \) are perpendicular to each other. This is because a zero dot product means that the cosine of the angle between them is zero, which occurs at 90 degrees.

Key concepts

Velocity Vector

A velocity vector provides us a comprehensive description of a particle's motion. It combines both the speed and direction of the particle. Imagine a car traveling down a road. Its speed might be 60 mph, but if we want to truly understand where it's headed, we need a direction as well - that's the essence of a velocity vector.

Mathematically, a velocity vector is written as \( \mathbf{v}(t) \), where \( t \) represents time. The vector consists of components along the x, y, and possibly z axes. For example, if a particle moves in a two-dimensional plane, its velocity vector might look like \( \mathbf{v}(t) = (v_x(t), v_y(t)) \).

• The magnitude of this velocity vector is what we commonly refer to as speed.
• If the speed is constant, it means the particle's path is straight and steady in terms of velocity magnitude but could curve in terms of direction.

Understanding the velocity vector is crucial in determining how an object navigates through space.

Acceleration Vector

The acceleration vector represents the rate at which a particle's velocity changes over time. Just as the velocity vector tells us the speed and direction of movement, acceleration tells us how this speed and direction evolve.

In physics, acceleration is denoted as \( \mathbf{a}(t) \). If you imagine driving a car, acceleration is similar to how you would press harder on the gas pedal to go faster, or ease off to slow down. It can change both how fast you're going and where you're going.

• An acceleration vector can cause a change in magnitude (speed) or direction of the velocity vector.
• If the speed is constant, any change caused by acceleration must happen perpendicularly to the direction of motion. This is what keeps the speed uniform while allowing direction change.

Therefore, understanding acceleration is key to figuring out how and why a particle's movement shifts.

Dot Product

The dot product is a fascinating operation between two vectors. It helps us determine the angle-like relationship between them. When two vectors are multiplied using the dot product, the result is a scalar, not a vector.

The dot product between two vectors \( \mathbf{u} \) and \( \mathbf{v} \) is denoted as \( \mathbf{u} \cdot \mathbf{v} \). This operation is fundamentally understood as \( |\mathbf{u}| |\mathbf{v}| \cos{\theta} \), where \( \theta \) is the angle between the two vectors.

• The significance of the dot product being zero is profound - it implies the vectors are perpendicular or orthogonal.
• If velocity and acceleration vectors are perpendicular, their dot product results in zero, confirming constancy of speed.

This numeric result can thus provide crucial geometrical insights without directly measuring angles, simplifying many physics problems.

Perpendicular Vectors

Perpendicular vectors are fascinating because they intersect at right angles to each other. This alignment has a unique significance in vector mathematics, especially when dealing with movement and forces.

In our context, when a velocity vector and an acceleration vector are perpendicular, it implies there's no change in the speed magnification. The angle between them is 90 degrees.

• When the angle \( \theta \) between two vectors results in the dot product \( \cos{\theta} = 0 \), they are perpendicular.
• For a moving object with constant speed, its acceleration causes changes solely in direction, not in speed magnitude.

Understanding this perpendicular relationship is vital in physics and engineering as it forms the basis for many real-world applications.