Notes
Apollo 15: Hammer and Feather demonstration: http://video.google.com/videoplay?docid=6926891572259784994#
Some Terminology and Definitions
Mechanics: The study of the motion of objects, and the related concepts of force and energy. Has two sub categories
Ø Kinematics: Description of how objects move.
Ø Dynamics: Deals with forces and why objects move as they do (be careful with using the word why)
Translational Motion: Motion without rotation. One-dimensional or straight line.
¢ Particle: A mathematical point that has no spacial extent or size.
¢ Reference Frame: Frame of reference - what quantities are being measured or compared with respect to.
Einstein created some interesting thought experiments dealing with reference frames - covered later.
¢ Coordinate Axes: Useful in representing a frame of reference - 2 lines drawn at 90 degree angle that divide the x-y plane. Used to specify the
position of an object.
¢ Displacement: How far an object has moved from its starting point
¢ Distance: Length of the total path travelled
¢ Vector: Operations with will be covered later - displacement with direction specified
¢ Scalar: Has magnitude or numerical value only
¢ Average Speed = s = distance travelled/time elapsed = (x2 - x1)/(t2 - t1) or Dx/Dt. Expressed in units of distance per time - it is a scalar quantity.
Where x2 is final position, x1 is initial position, t2 is final time, t1 is initial time.
Note that this is a scalar quantity; no direction is specified
¢ Average Velocity = v = displacement travelled/time elapsed.
Note that this is a vector quantity because direction is specified
¢ Instantaneous Velocity: v = lim Dx/Dt
Dt ]0
Dt ]0 means as the change in time approaches 0. Note: This is as close to calculus as we will get (required) in this course.
Unless stated otherwise, references to velocity will mean instantaneous velocity.
¢ Instantaneous Speed. s is always equal to the magnitude of the instantaneous velocity. Why?
¢ Acceleration: Occurs when velocity is changing. It specifies how rapidly the velocity of an object is changing.
¢ Average Acceleration: a = change of velocity/time elapsed = (v2 - v1)/(t2 - t1) = Dv/Dt, where the symbols have the meanings as stated above.
Acceleration is a vector quantity but, for one-dimensional motion, we will use only + or - to indicate direction reference the reference frame.
¢ Instantaneous Acceleration: a = lim Dv/Dt
Dt ]0
Motion at Constant Acceleration
¢ From the above definition of average acceleration
a = change of velocity/time elapsed = (v - v0)/(t - t0)
Note that I am using v as the final velocity, v0 as the initial position, t as the final time, and t0 as the initial time. I will use this approach throughout the course.
But, since we are, for now, assuming constant acceleration, a = a. Additionally, since we start at time 0, this becomes
a = (v - v0)/t
This (cross multiplying, etc) can be solved for v to give
v = v0 + at First Useful Relationship Question: What is being said, in plain English?
In words: the velocity at any time t is equal to the initial velocity + acceleration (constant) times elapsed time
¢ From the above definition of average velocity
v = (x - x0)/(t - t0).
If we set t0 as 0 (we start at time 0), then
v = (x - x0)/t
This (cross multiplying, etc) can be solved for x to give
x = x0 + vt
Because the velocity increases at a uniform rate (consequence of constant acceleration), v will be midway between the initial and final velocities or
v = (v0 + v)/2. Second Useful Equation Question: What is being said, in plain English?
We can now combine this equation with the following 2 equations that were derived above
v = v0 + at
x = x0 + vt
to give
x = x0 + vt = x0 + ((v0 + v)/2)t or
x = x0 + ((v0 + v0 + at)/2)t or
x = x0 + vot + (1/2)at2 or, equivalently, (we are in 1-D)
y = y0 + vot + (1/2)at2 Third Useful Equation Question: What is being said, in plain English?
Rearranging these 3 equations, we can derive a fourth useful equation for situations when time t is not known
v2 = v02 + 2a(x -x0) Fourth Useful Equation Question: What is being said, in plain English?
Summary
The four equations discussed above and repeated below are referred to as the Kinematic Equations for Constant Acceleration
| v = v0 + at | |
| v = (v0 + v)/2. | |
| x = x0 + vot + (1/2)at2 or, equivalently, y = y0 + vot + (1/2)at2 | |
| v2 = v02 + 2a(x -x0) |
¢ Galileo: Father of modern science
At a given location on Earth and in the absence of air resistance, all objects fall with the same constant acceleration.
¢ g = acceleration due to gravity = 9.80 m/s2
¢ g is a vector; its direction at any time is towards the center of the Earth
¢ Notes
ð Acceleration and velocity are not always in the same direction
ð An object thrown upward does not have zero acceleration at its highest point
ð If up is positive then g has a negative sign
ð If up is down then g has a positive sign
¢ Paper and pen drop experiment
