s= mt+constant where m=Ds/Dt i.e., speed When plotting distance (s) versus time (t) the slope is simply the speed (magnitude of the velocity). s Dt Ds t s= mt+constant where m=Ds/Dt i.e., speed

What if the motion is more complicated than constant velocity What if the motion is more complicated than constant velocity? How do we determine the instantaneous velocity then? We can get closer to the instantaneous speed by making the path segment smaller and smaller… s t

Or we can get closer and closer by making Dt smaller and smaller… * Dt Ds * Dt t

making Dt smaller and smaller is “taking the limit” We now call that “taking the derivative”

Now the integrals or inverse derivatives can be computed so we can start with the sum of all the forces acting on a body and figure it backward until we get to the position and velocity as a function of time.

We get these equations for constant acceleration motion All from adding up all the forces acting on a body!

The angle measured by going completely around a circle is 360o But that is pretty arbitrary – why not tie it to something that is meaningful?

Lets remember some high school geometry: q x In most astronomical observations, the x or h lengths are large compared to the y size scale and in order to observe features of size y in an object the angular resolution (q) must be as small as possible.

If a function has continuous derivatives up to (n+1)th order, then this function can be expanded in the following fashion: If the Rn term (remainder) converges to a value, this is the Taylor series expansion of the function f.

r s q r

Angular Momentum angular momentum – the momentum involved in spinning /circling = massnvelocitynradius torque – anything that can cause a change in an object’s angular momentum (twisting force) massnaccelerationnradius

v m r a = v2/r F = m v2/r

Who cares about momentum (mv)? Conservation of linear Momentum In the absence of a net force, the total linear momentum of a system remains constant. But this is just Newton’s first law!! mv = constant

Conservation of Angular Momentum In the absence of a net torque, the total angular momentum of a system remains constant.

Universal Law of Gravitation Between every two objects there is an attractive force, the magnitude of which is directly proportional to the mass of each object and inversely proportional to the square of the distance between the centers of the objects. M1M2 d2 Fg a

G = 6.67 x 10-11 N m2/kg2 M1M2 d2 Fg = G

2pR F = G R m m R v = distance / time v = 2pR / P F = a = = v / R G R circumference = 2pR F = G R 2 m 1 m 2 1 R v = distance / time v = 2pR / P F = a = = v / R 2 G R m 1 F = a = v / R 2 m 1 F = a m 1 v = G /R m 2 4p R / P = G /R 2 m 4p R = G P 2 m 3 P2 = (4p2/Gm2) R3

Energy units large and small What is conservation of Energy? What does Einstein’s mass energy equation mean? What does an electron do? What does a force do? What does mean for an arbitrary atom? What is acceleration? What are Kepler’s three laws? What are Newton’s three laws?