Velocity and Center of Mass
What happens if a rocket launches up into the air, then explodes? Where is the center of Mass? What about the velocities? Well, The velocity of the center of mass remains the same. But for instance, if you launch a rocket with two stages, and at the peak of the parabola, jettison the spent fuel canister? It’s velocity becomes zero, as the second stage kicks in and pushes off from the first stage. It picks up double the velocity (if they are equal weight, and is able to fly twice as far as the two masses would have gone together. The first stage falls straight down.
CIRCULAR MOTION Chapter 10
To keep track of Circular motion, we need to watch the same things we watch for Linear motion.
- Position r (letter r)
- Velocity ω (little omega)
- Acceleration α (little alpha)
But now we are moving into Polar Coordinates, where it is much easier to keep track of a position on a circle in terms of radius and theta θ (Angle from the origin). If you use Cartesian coordinates you have to write x and y as r cos θ and r sin θ. So it’s easier to use Cartesian. Theta is also now always expressed in radians, or “length of arc over radius”
What we do with the Polar Coordinates:
Theta in Radians equals length of arc over radius
1 ) Average Angular Velocity (little omega) equals change in Theta over change in Time.
2 ) Instantaneous Angular Velocity(little omega) equals the derivative of theta with respect to the derivative of time.
3 ) The derivative of Theta with respect to time is equivalent to the derivative of length with 4) respect to time multiplied by one over the radius.
5 ) The Tangential velocity at a time is equal to the derivative of the length with respect to time.
6 ) The Average Angular Velocity (little omega) equals the tangential velocity over the radius.
7 ) The Average Angular Acceleration (little alpha) is equal to the change in Velocity (omega) over the change in time.
8 ) The angular instantaneous acceleration (little alpha) is equal to the derivative of angular velocity with respect to time.
You can use the same formula’s used for Linear constant acceleration, just replace the Linear velocity and acceleration with the Circular velocity and acceleration.