# Velocity and Acceleration Vector problem

• Albeaver
In summary, the car moves on a circle with a constant radius and its speed changes with time according to the equation v = ct. The direction of the velocity and acceleration vectors can be found by drawing a diagram and solving for the velocity and acceleration vectors. The angle between the velocity vector and the acceleration vector can be found by using the Pythagorean theorem.
Albeaver

## Homework Statement

A car moves on a circle of constant radius b. The speed of the car varies with time according to the equation, v = ct, where c is a positive constant.
a) Draw a diagram showing the direction of the velocity and acceleration(s). Find the velocity and acceleration vectors (Directions of the vectors you have chosen to show in your diagram).
b)Find the angle between the velocity vector and the acceleration vector. (Note: Express the angle in terms of c and t)

V = dx/dt
A = dv/dt

## The Attempt at a Solution

Position Vector (from center of circle): b cos (u(t))i +b sin(u(t))j;
u(t) = a function of time
Velocity vector: -b ucos(u(t))i + b u sin(u(t))j;
bu(t) = ct
u(t) = 1/2 (c/b)t^2

Velocity Vector: -(c)(t)sin(1/2(c/b)t^2)i+(c)(t)cos(1/2(c/b)t^2)j
Acceleration Vector: (c-(c^2 t^2)/b)cos(1/2(c/b)t^2)i+((-c^2 t^2)/b-c)sin(1/2(c/b)t^2)j

I'm not sure if I did this correct. If not can you please show me my error and help with part b? :)

Albeaver said:
Position Vector (from center of circle): b cos (u(t))i +b sin(u(t))j;
u(t) = a function of time
Velocity vector: -b ucos(u(t))i + b u sin(u(t))j;
That differentiation is wrong - try it again.

Thanks...
Velocity vector: b usin(u(t))i - b u cos(u(t))j;

Albeaver said:
Thanks...
Velocity vector: b usin(u(t))i - b u` cos(u(t))j;

You've corrected the trig functions but now the signs are wrong.
For part (b), given two vectors, how do you find the angle between them?

Oh yeah I forgot to put that didn't I? Cos (theta) = (v dot a)/(|v||a|) Is that correct?

## 1. What is the difference between velocity and acceleration?

Velocity is the rate of change of an object's position with respect to time, while acceleration is the rate of change of an object's velocity with respect to time. In simpler terms, velocity tells us how fast an object is moving, while acceleration tells us how much an object's velocity is changing.

## 2. How do you calculate velocity and acceleration vectors?

To calculate the velocity vector, you need to divide the change in displacement by the change in time. The result will be a vector with both magnitude and direction. Acceleration is calculated by dividing the change in velocity by the change in time. It is also a vector with both magnitude and direction.

## 3. How do you represent velocity and acceleration vectors graphically?

Velocity and acceleration vectors are usually represented graphically using arrows. The length of the arrow represents the magnitude of the vector, and the direction of the arrow represents the direction of the vector. The orientation of the arrow can be determined by using the right-hand rule.

## 4. What is the difference between average and instantaneous velocity and acceleration?

Average velocity and acceleration are calculated over a period of time, while instantaneous velocity and acceleration are calculated at a specific moment in time. Average velocity and acceleration give an overall picture of an object's motion, while instantaneous velocity and acceleration give a more detailed view.

## 5. How are velocity and acceleration vectors used in real-life applications?

Velocity and acceleration vectors are used in various fields such as physics, engineering, and sports. In physics, they are used to analyze the motion of objects. In engineering, they are used to design structures and machines with specific motion characteristics. In sports, they are used to analyze athletes' performances and improve their techniques.

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