Velocity of point rotating on a wheel as the wheel moves

[Psip]

Homework Statement

So I have to prove that the total velocity of a particle on a wheel is Vcm+Ui=Vi where Ui is the particles velocity if the wheel's center of mass wasn't moving so just v=omega*R I'm assuming and Vcm is the velocity of the center of mass and Vi is the total velocity of the point on the wheel.

Homework Equations

a^2+b^2=c^2

The Attempt at a Solution

You can draw a diagram of the before and after of the particle after time t. You can then place the x and y components for the Ui and Vcm head to tail and you can get the (Uix+Vcm)^2+(Uiy)^2 = Vi^2 (Vcm only has an x component) but I don't know how to get from this to Vcm+Ui=Vi

 

[ehild]

You mix speed and velocity. Velocity is vector, speed is scalar. The velocity of a particle on the wheel is not omega * R. It is its speed with respect to the centre of the wheel.
You can not prove that the velocity of the particle with respect to the ground is the sum of the velocity of CM of the wheel and the velocity of the particle with respect to the CM. It is he basic assumption of the Classical Mechanics, derived from experience. The velocities add up as vectors. You can write
$\vec V_i=\vec V_{cm} + \vec U_i$, but it is not true for the magnitudes, V_i U_i and V_cm.

 

[Psip]

Yes that's what I mean I just didn't know how to put the vector arrow above the letters but yes they are all vectors. I just don't know how to prove the equation you listed based on a picture of the before and after of the point. You can make a displacement vector from the point to the same point on the wheel in the future as if it hadn't rotated thus making a straight line. Then you can make a displacement vector from that point to a little down and to the right to represent the displacement if that wheel just moved forward without rotating then stopped and rotated which would be the same as if it had moved and rotated at the same time.

 

[ehild]

Do the same with the velocities as with the displacements. The velocities add in the same way as the displacements do. It is vector addition. No need to prove anything.

 

[Psip]

But how do I know that the total displacement is the addition of the two displacements because the 2 sides of a triangle added should be longer than the longest side which is why I'm confused.

Edit: I understand now because the displacement vectors are added which gives total displacement

 

[ehild]

When you add two vectors you place the tail of the second one at the head of the first one and the resultant is the vector that points from the free tail to the free head. They make a triangle, and you know, that the sum of the lengths of two sides is always greater than the third side.