# Motion in Space: Velocity and Acceleration

• _N3WTON_
In summary, the conversation discusses the position functions of objects A and B, which describe different motion along the same path. The paths are traced out in a counterclockwise direction and have a radius of 1. The velocity and acceleration of A and B are found and compared, with B having a larger normal component and therefore changing direction more quickly. The definitions of velocity and acceleration, in terms of position and velocity respectively, were used incorrectly but have been corrected.
_N3WTON_

## Homework Statement

The position function of objects A and B describe different motion along the same path for t >= 0.
A: $r(t) = cos(t)i + sin(t)j$
B: $r(t) = cos(3t)i + sin(3t)j$
a) Sketch the path followed by A and B
b) Find the velocity and acceleration of A and B and discuss the distance
c) Express the acceleration of A and B in terms of the tangential and normal components and discuss the difference

## Homework Equations

T(t) = velocity/speed (unit tangent vector)
N(t) = T'(t)/ absolute value of T'(t) (unit normal vector)
$a_T = v'$
$a_N = kv^{2}$
$a = v'T + kV^{2}N$ (k is curvature)

## The Attempt at a Solution

Ok, this was a problem for my multivariable calculus class but if the mods feel that it belongs in a physics subforum then by all means move it there :).
a) I know that both functions are circles with radius r = 1, traced out in a counterclockwise direction
b) $v(t)_A = \int r(t) dt = -sin(t)i + cos(t)j$
$v(t)_B = \int r(t) dt = -3sin(3t)i + 3cos(3t)j$
$a(t)_A = \int v(t) dt = -cos(t)i - sin(t)j$
$a(t)_B = \int v(t) dt = -9cos(3t)i - 9sin(3t)j$
The difference between the two is that the 'B' function is traced out much faster than the 'A' function
c) For A:
$T(t) = -sin(t)i + cos(t)j$
$N(t) = -cos(t)i - sin(t)j$
The magnitude of T, N is equal to 1, as is the curvature k
Tangential acceleration is:
$a_T = -cos(t)i - sin(t)j$
Normal acceleration is:
$a_N = sin^2(t)i + cos^2(t)j$
For B:
$T(t) = -sin(3t)i + cos(3t)j$
$N(t) = -cos(3t)i - sin(3t)j$
The magnitude of T, N is equal to 3 and the curvature k is equal to 1
$a_T = -9cos(3t)i - 9sin(3t)j$
$a_N = 9sin^2(3t)i + 9cos^2(3t)j$
The tangential component shows the rate of change of velocity and the normal component shows the rate of change of direction. In this case the tangential components of the two functions are equal. However B has a larger normal component, therefore it changes direction more quickly. This problem was worth 20 points; however, I only received 16... I am allowed to make a correction for full credit so I was hoping somebody could tell me where I went wrong :)

What's the definition of velocity, in terms of position? What's the definition of acceleration, in terms of velocity?

What you wrote for the answers to the velocity and acceleration are OK, but the definitions you used are incorrect.

SteamKing said:
What's the definition of velocity, in terms of position? What's the definition of acceleration, in terms of velocity?

What you wrote for the answers to the velocity and acceleration are OK, but the definitions you used are incorrect.
Thank you for pointing that out, I'll make the changes in a moment...know that when I was finding velocity and accel I did differentiate. I did not integrate

I can't seem to edit my original post, perhaps the mods would be so kind as to change my mistake :)

SteamKing said:
What's the definition of velocity, in terms of position? What's the definition of acceleration, in terms of velocity?

What you wrote for the answers to the velocity and acceleration are OK, but the definitions you used are incorrect.
Also, besides the definition mistake was there anything else you noticed that was wrong? I have double checked but I believe that I answered everything correctly

## 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.

## How is velocity calculated?

Velocity is calculated by dividing the change in an object's displacement by the change in time. This can also be represented as the derivative of an object's position with respect to time.

## What is acceleration?

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

## How is acceleration related to velocity?

Acceleration and velocity are closely related. If an object's velocity is changing, then it is accelerating. The magnitude of the acceleration is directly proportional to the rate of change of velocity.

## Can an object have a constant velocity and still accelerate?

No, if an object has a constant velocity, it means that its speed and direction are not changing. In order for acceleration to occur, there must be a change in velocity, either in magnitude or direction.

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