Understanding Velocity and Acceleration of a Moving Particle

In summary, the conversation discusses the position and velocity of a particle described by the function $\vec{r}=\cos{(\omega t)}\hat{i} + \sin{(\omega t)}\hat{j}$ and how to show that the velocity vector is perpendicular to the position vector and that their dot product is always zero. It is suggested to use Calculus to find the derivatives and the dot product to show their perpendicularity.
  • #1
WMDhamnekar
MHB
376
28
A particle moves so that its position vector is given by $\vec{r}=\cos{(\omega t)}\hat{i} + \sin{(\omega t)}\hat{j}$. Show that the velocity $\vec{v}$ of the particle is perpendicular to $\vec{r}$ and $\vec{r} \times \vec{v}$ is a constant vector.

How to answer this question?
 
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  • #2
First calculate \(\displaystyle \vec{v} = \dfrac{d \vec{r}}{dt}\). If \(\displaystyle \vec{r} \cdot \vec{v} = 0\) for all t then they are always perpendicular.

-Dan
 
  • #3
Since the velocity function is the derivative of the position function and the acceleration function is the derivative of the velocity function, I would say, "start by taking a Calculus class!". Have you done that? Do you know what the derivatives of $cos(\omega t)$ and $sin(\omega t)$ are? Do you know how to show that one vector is perpendicular to another? (Dot product.)
 

Related to Understanding Velocity and Acceleration of a Moving Particle

What is velocity?

Velocity is a measure of an object's speed and direction of motion. It is a vector quantity, meaning it has both magnitude (speed) and direction.

What is acceleration?

Acceleration is the rate at which an object's velocity changes over time. It is also a vector quantity, with both magnitude (how much the velocity changes) and direction.

How are velocity and acceleration related?

Velocity and acceleration are related in that acceleration is the rate of change of velocity. This means that when an object's velocity changes, it is accelerating. However, an object can have a constant velocity and still be accelerating if its direction of motion changes.

What is the difference between average and instantaneous velocity?

Average velocity is the total displacement of an object divided by the total time taken, while instantaneous velocity is the velocity of an object at a specific moment in time. Average velocity gives an overall idea of an object's motion, while instantaneous velocity gives a more precise understanding at a specific point in time.

How can we calculate acceleration?

Acceleration can be calculated by dividing the change in velocity by the change in time. This can be represented by the equation a = (vf - vi) / t, where a is acceleration, vf is final velocity, vi is initial velocity, and t is time. Acceleration can also be calculated using the slope of a velocity-time graph.

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