Velocity as a function of radial distance on an elliptical trajectory

[psid]

Homework Statement

Assume that a point on an ellipse is described by the vector $$r=m(a\cos{\theta},b\sin{\theta})$$, where $$0\leq m\leq 1$$ and that the vector is rotating in the clockwise direction at constant tangential velocity $$W$$ when $$m=1$$.

The problem is to find the velocity $$W(m)$$.

Also, is the angular velocity constant as it is for circular motion?

 

[nickjer]

Did you take the time derivative of it and then calculate the magnitude of dr/dt?

 

[nlightner]

I would approach this problem by first understanding the concept of velocity and its relation to radial distance on an elliptical trajectory. Velocity is defined as the rate of change of position, or in other words, how fast an object is moving and in what direction. In this case, the velocity is a function of the radial distance, which means that it changes as the object moves along the ellipse.

To find the velocity, we can use the given vector r=m(a\cos{\theta},b\sin{\theta}) and apply the formula for velocity, which is V=dr/dt, where dr is the change in position and dt is the change in time. In this case, we can rewrite the vector as r=m(a\cos{\theta},b\sin{\theta})=ma(\cos{\theta},\sin{\theta})=ma\hat{r}, where \hat{r} is the unit vector in the radial direction. This means that the change in position, dr, is equal to ma\hat{r}.

Next, we need to find the change in time, dt. Since the object is rotating in a clockwise direction at constant tangential velocity W, we can use the formula for angular velocity, which is \omega=d\theta/dt, where \omega is the angular velocity and d\theta is the change in angle. We know that when m=1, the vector is rotating at this constant tangential velocity, so we can write the equation as W=\omega d\theta/dt. Solving for dt, we get dt=d\theta/W.

Now, we can substitute this value for dt into our formula for velocity, V=dr/dt, and get V=ma\hat{r}/(d\theta/W). Simplifying, we get V=ma\hat{r}W/d\theta. Since m is a constant and a\hat{r} represents the radial distance, we can rewrite this as V=arW/d\theta.

Therefore, we can see that the velocity, V, is indeed a function of the radial distance, r, and the angular velocity, W. As m decreases, the radial distance decreases, and the velocity also decreases. This makes sense intuitively, as the object would be moving slower as it gets closer to the center of the ellipse.

To answer the second part of the question, the angular velocity is not constant in this