# The unit tangent/normal vectors to motion+their derivatives

• -Vitaly-
In summary, the homework statement is trying to solve for the magnitude of n, u=1, u is in the direction of the velocity u=V/V, and the attempt at a solution is having trouble with the second part. The alternate approach using theta will work and the last step is to find the relation between ds/dt and v. The relation between ds/dt and v is not known, but they are equal.
-Vitaly-

## Homework Statement

http://img124.imageshack.us/img124/4485/clipboard01hb6.jpg

## Homework Equations

magnitude of n, u=1, u is in the direction of the velocity u=V/V

## The Attempt at a Solution

The 1st part is easy, I wrote:
http://img129.imageshack.us/img129/1244/clipboard01mv3.jpg

But I can't do the second part, I read about 10 different sources about these unit vectors, but now I'm even more confused. Especially about so called "radial" and "transverse" unit vectors, they are not "tangential" and "normal" unit vectors?

Anyway, for this part I tried analytical approach:
http://img129.imageshack.us/img129/7342/clipboard01tu6.jpg
But not sure what to do next. Or another approach:
http://img129.imageshack.us/img129/287/clipboard02ve7.jpg

Any help will be appreciated, I spent about 2 days on this problem alone

Last edited by a moderator:
Use the chain rule to express $\math d \hat u / dt$ in terms of $\math d \hat u / ds$.

Your alternate approach using theta will work. Let

$$\rho\mbox{d\theta}=ds$$

Integrate with respect time. V(s) = ds/dt and

$$\rho\frac{d\theta}{dt}=\rho\omega$$

and

$$\omega{ds}= \mbox{magnitude of V}$$

ok, but I don't know what to do next :(

$$\frac{d \hat u}{ds}=\frac{d \hat u}{ds}$$$$\frac{ds}{dt}$$

You know $$\frac{d\hat u}{ds}$$ from part (a), and you should know $$\frac{ds}{dt}$$.

I still can't complete it :(
http://img89.imageshack.us/img89/2286/clipboard01da6.jpg

The last step, why du/d(theta)=n?

Last edited by a moderator:
Express the unit vectors n and u in terms of unit vectors x and y using cosine and sine. The unit vectors x and y do not change direction. Now, take du/d(theta) and this will show how this derivative is related to n.

What is the relation between ds/dt and v?

D H said:
What is the relation between ds/dt and v?
Are they equal? I don't know, this is too hard :( please just tell me how to do it, so I know for future.

## 1. What is the unit tangent vector to motion?

The unit tangent vector to motion is a vector that represents the direction of motion at a specific point on a curve. It is a unit vector, meaning it has a magnitude of 1, and is tangent to the curve at that point.

## 2. How is the unit tangent vector calculated?

The unit tangent vector is calculated by taking the derivative of the position vector with respect to time and then dividing it by its magnitude. This can also be written as the velocity vector divided by its magnitude.

## 3. What is the significance of the unit tangent vector in motion?

The unit tangent vector is important because it represents the direction of motion at a specific point on a curve. It can also be used to calculate the curvature of the curve at that point, which is important in understanding the acceleration of an object moving along the curve.

## 4. What is the unit normal vector to motion?

The unit normal vector to motion is a vector that is perpendicular to the unit tangent vector and points towards the center of curvature of the curve at a specific point. It is also a unit vector, meaning it has a magnitude of 1.

## 5. How is the unit normal vector related to the unit tangent vector?

The unit normal vector is always perpendicular to the unit tangent vector, meaning they form a right angle. They both play important roles in understanding the motion of an object along a curve, with the unit normal vector representing the direction of acceleration towards the center of curvature.

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