Question

I have to give three calculus-based introductory physics describing motion.
are these right?

Answer 1

Constant Force F-- e.g., motion of an object falling a few meters near the surface of the Earth (in which case the constant force depends on the particle's mass: F=-mg, resulting in all falling objects having the same (downward) acceleration: g=9.8 m/s2)
a=F/m=-g: acceleration is the result of applying the force; it can be calculated by the force divided by the particle's mass

z(t)=z0+v0t+½at2: the height of the object (z) depends on the initial height (z0), the initial velocity (v0) and time (t)

v(t)=v0+at: the velocity changes uniformly in time from its initial value

U(z)=-Fz=mgz: the potential energy has the property that if you take minus the derivative of it w.r.t. position, you get the force. -Fz + constant works; we've set the constant equal to zero.

In the case of the ball falling near the surface of the Earth, the above described motion cannot continue indefinitely as the ball soon encounters the ground. In the case of a perfectly elastic collision with the ground, the ground provides a force to exactly reverse the ball's velocity. It bounces forever between the ground (z=0) and some maximum height (zmax) that depends on its energy.

Spring Force (Hooke's Law) F=-kx -- e.g., a force that always pulls the object back to the equilibrium position (x=0)...if x>0, the force is in the negative x direction, etc.
x(t)=A sin(t+): the particle oscillates around equilibrium getting as far away as ±A. The period, T (the time it takes to make one complete oscillation), is determined by (the angular frequency): T=2/. is in turn determined by the strength of the spring, k (called the spring constant) and the mass m: 2=k/m. Thus a strong spring connected to a light particle will oscillate quickly, i.e., with a short period.

v(t)=A cos(t+): the velocity (v) of the particle also oscillates, i.e., sometimes the particles is moving to the right (positive v) sometimes it is moving to the left (negative v). Notice that the particle has is maximum speed (of A) when the cosine term reaches its extremes of ±1. That happens only when sine is zero (because cos2+sin2=1) and hence the particle is moving through the equilibrium position (x=0). Similarly, the particle is momentarily at rest (v=0) only when the particle is at an extreme position (±A; i.e., if cosine is zero, sine must be ±1).

U(x)=½kx2: the potential energy has the property that if you take minus the derivative of it w.r.t. position, you get the force. ½kx2 + constant works; we've set the constant equal to zero.

Orbital Motion: motion of a particle with a /r2 central force applied. In intro physics the topic was the motion of planets under the influence of the Sun's gravitational force...orbital mechanics. In quantum mechanics the topic is the motion of an electron under the influence of the electrostatic attractive force of the nucleus...atomic physics. Equivalent equations for the force (and hence "the same" as far a physics goes) but quite different distance scales. In addition energy emission by gravitational radiation is totally negligible in the Solar System, whereas energy emission by electromagnetic radiation (light) should be important in an atom.
Planets move in ellipses with the Sun at one focus. Ellipses can be described in terms of their semi-major axis, a, (basically the longest radius) and their eccentricity (basically how squashed the ellipse is: e=0 is a circle, a fully squashed ellipse looks like a line and has e=1).

r=a(1-e2)/(1+e cos()); rmin= a(1-e); rmax=a(1+e)

, the polar angle from closest approach is given the odd name: true anomaly. The timing of the motion (i.e., when the planet or electron has a particular ) is a bit complex. The game is to express the true anomaly () in terms of the eccentric anomaly (u) and then find an expression relating time and the eccentric anomaly. For nearly circular orbits it turns out that the true anomaly, the eccentric anomaly, and the mean anomaly (t) are all approximately equal to each other. I apologize for this archaic nomenclature, but physics is stuck with these names. Below find the geometric construction that relates the true anomaly and the eccentric anomaly, the formula relating these two, and the formula relating time and the eccentric anomaly.

Answer 2

i will post what the queston marks represent

Answer 3

@iGreen @Zarkon @ParthKohli

Answer 4

@Conqueror @bibby

Answer 5

@iGreen

Answer 6

O_O

I have no idea..

Answer 7

is this too much for you?

Answer 8

i wrote it myself :D