Question

A particle is moving in the \(x y\) plane with velocity \(\overrightarrow{\mathbf{v}}(t)=\) \(v_{x}(t) \hat{\mathbf{i}}+v_{y}(t) \hat{\mathbf{j}}\) and acceleration \(\mathbf{a}(t)=a_{x}(t) \hat{\mathbf{i}}+a_{y}(t) \hat{\mathbf{j}} . \mathbf{B y}\) taking the appropriate derivative, show that the magnitude of \(\overrightarrow{\mathbf{v}}\) can be constant only if \(a_{x} v_{x}+a_{y} v_{y}=0\)

Short answer

The magnitude of the velocity vector is constant only if \(a_{x} v_{x}+a_{y} v_{y}=0\).

Step-by-step solution

Express the magnitude of velocity vector

The magnitude of the velocity vector \(\overrightarrow{\mathbf{v}}\) is given by \(|\overrightarrow{\mathbf{v}}| = \sqrt{v_x^{2} + v_y^{2}}\).

Differentiate the magnitude of velocity vector

If the magnitude of the velocity vector is constant, its derivative with respect to time, \(t\), should be zero. So, \(\frac{d|\overrightarrow{\mathbf{v}}|}{dt} = 0\). Computing the derivative we find \(\frac{d|\overrightarrow{\mathbf{v}}|}{dt} = \frac{v_x a_x + v_y a_y}{\sqrt{v_x^{2} + v_y^{2}}}\).

Show the condition for the magnitude of velocity to be constant

As stated in step 2, for the magnitude of the velocity to be constant, \(\frac{d|\overrightarrow{\mathbf{v}}|}{dt} = 0\). So, after rearranging, we find \(v_x a_x + v_y a_y = 0\).

Key concepts

Velocity Vector

In the realm of physics, understanding the concept of a 'velocity vector' is paramount for analyzing particle motion. A velocity vector, represented symbolically as \( \overrightarrow{\mathbf{v}} \), provides information about the speed and direction of a particle's movement. In a two-dimensional space such as the XY-plane, it can be expressed in terms of its components along the X and Y axes using unit vectors \( \hat{\mathbf{i}} \) and \( \hat{\mathbf{j}} \) respectively. So, if a particle's velocity varies over time, we denote this as \( \overrightarrow{\mathbf{v}}(t)= v_{x}(t) \hat{\mathbf{i}} + v_{y}(t) \hat{\mathbf{j}} \).
To fully comprehend particle motion, one must not only understand how to represent the velocity vector but also how to interpret its changes over time. This vector's magnitude and direction at any given moment can inform us about the trajectory of the particle's movement.

Acceleration Vector

Similarly crucial in dissecting particle dynamics is the 'acceleration vector'. Just like velocity, acceleration is also a vector quantity, indicated by \( \mathbf{a}(t) \). It illustrates how the velocity of a particle changes with time. In two dimensions, it is described by its X and Y components: \( \mathbf{a}(t) = a_{x}(t) \hat{\mathbf{i}} + a_{y}(t) \hat{\mathbf{j}} \).
Acceleration is the rate at which the velocity vector changes, and it is essential for understanding how an object speeds up, slows down, or changes its path of travel. When analyzing particle motion, it's important to note that the velocity vector can maintain a constant magnitude even when there's acceleration—provided this acceleration is perpendicular to the velocity. This concept is pivotal when explaining phenomena such as uniform circular motion.

Time Derivative

The 'time derivative' is a fundamental concept in physics, representing how a quantity changes instantaneously with respect to time. Mathematically, it is expressed as the derivative of a function in terms of the variable 't' for time. For instance, if we have a velocity function \( \overrightarrow{\mathbf{v}}(t) \), the time derivative of its magnitude is symbolized as \( \frac{d|\overrightarrow{\mathbf{v}}|}{dt} \).
In particle motion analysis, the time derivative becomes a powerful tool. By taking the derivative of velocity with respect to time, we essentially obtain the acceleration vector. If the velocity vector changes consistently over time, the derivative will provide us with precise information regarding the rate and direction of this change. Understanding time derivatives is crucial for solving many problems in physics, including determining whether a particle's speed remains constant over time.

Magnitude of Velocity

The 'magnitude of velocity', often just referred to as speed, is a scalar value representing how fast a particle is moving regardless of its direction. For a velocity vector \( \overrightarrow{\mathbf{v}} \), the magnitude is calculated using the Pythagorean theorem as \( |\overrightarrow{\mathbf{v}}| = \sqrt{v_x^{2} + v_y^{2}} \).
When we say the magnitude of velocity is constant, it means that the speed of the particle isn't changing over time. This can be the case even if the particle's direction of motion is changing—picture a car moving at a steady speed around a circular track. The magnitude of its velocity (its speed) remains constant, while the direction (velocity vector) is continually changing. To determine if the magnitude of velocity is indeed constant, one would calculate the time derivative of the magnitude, and if this derivative equals zero, the speed is unchanging. This calculation is essential for confirming that the particle's movement adheres to certain physical conditions or constraints, such as moving with uniform velocity.