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.
We learned 4 equations to describe motion in one dimension with constant acceleration.
These Equations are:
- Position at time is equal to initial position plus initial velocity times time, plus one half acceleration times time squared.
- Ending velocity equals initial velocity plus acceleration times time.
- Average velocity equals initial velocity plus ending velocity divided by two.
- Ending velocity when time is unknown is velocity squared equals initial velocity squared plus two times acceleration times ending position minus initial position.
***NOTE: The image for C is wrong, it should have a 2 on the bottom not a t.
The Average Velocity between points x0 and x1 is equal to the instantaneous velocity at point x½
Acceleration due to gravity near the earth surface:g=-9.8m/s2
When an object is traveling at a constant velocity, it is special. It’s Instantaneous Velocity and Average Velocityare the same, and the position of the object can be stated by:
Read this: “The position of the object is at point x which is equal to the origin position plus Velocity times Time.”
The slope between two positions of the object tells us the average velocity. Slope can be in meters per second or seconds per meter, but it should have a unit attached to it, in the form of Unit per Unit. On a graph of velocity, time would occupy the x axis, and position would occupy the y axis.
Instantaneous Velocity can be thought of as constant velocity between two very close positions. So this starts to get into Calculus, which I need to review, but expressing Instantaneous Velocity is really just the same as average velocity as the x axis approaches 0. (Since x is Time, as the time is shorter the closer you are to an instant)
Acceleration tells you about how fast the velocity of an object is changing. There is also an Average and Instantaneous Acceleration. When speaking about acceleration, velocity is along the x axis and Time and along the y axis. The basic formula is the same as velocity, with the only difference the axis of the graphs.
The second half of the class the Professor spoke about some special equations for objects with a constant acceleration. You can use these to get the Average Velocity or The Position or Time, depending on what you have to work with!
x = position
v = velocity
t = time
a = Constant Acceleration
A note on Speed:
Speed is the same as velocity, but it does not tell about direction! A car going -30mph is traveling at a speed of 30 miles per hour somewhere, but a car at a velocity of -30mph is going in reverse at a speed of 30 miles per hour.