# The Ungerecht Family Tree

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### Newton's 3 laws

1) objects in motion stay in motion, a body at rest stays at rest, until a force is applied ("law of inertia")
2) change in momentum of a body is equal in magnitude and direction to the force applied to it (force = mass * acceleration)
3) when two bodies interact, they apply forces that are equal to each other, and opposite in direction ("law of action and reaction")

Basic definitions:

• Force is in newtons or pounds. One newton = 1kg * m / s² (the force needed to accelerate 1kg at 1 m/s²)
• f = ma
• Momentum is p
• p = mv
• energy = work (joules) = force * distance
• J = F*d = applying 1 newton for 1 meter (units of kg * m²/s²)
• F = J/d
• power = work/time (joules/sec or watts)

Fnet = Δp / Δt (since p = mv and Δv/Δtime = acceleration)

Change in potential energy is given by U=mgh

• potential energy:
• U = 1/2 kx² (spring), or
• P = mgh (at mass at some height, even on an inclined plane)
• kinetic energy: K = 1/2 mv²

dimensional homogeneity - units must be correct for parts added together, left side matches right side, etc.

### Distance, time, velocity, acceleration

Displacement is change in position.

s(t) = s0 + t*(v0+vt)/2
s = displacement from origin at time t
vt = v0 + a*t
if v0 = 0 then
s(t) = s0 + t²*(a)/2
so in free-fall, from position 0, you have:
s(t) = g * t²/2

### Collisions

• elastic: Two objects bounce off each other. Kinetic energy, momentum conserved, no other energy created
• inelastic Two objects stick to each other. Momentum conserved, kinetic energy is not conserved (some energy converted to heat, sound, etc.)

coefficient of restitution = ratio of energy conserved after collision

e = (vel. after collision) / (vel. before collision)
(for collision with immovable object)
e = (Vfa * Vfb) / (Via * Vib)
(for collision between objects a and b. f = final, i = initial velocity)
e = 1 for perfectly elastic, 0 for perfectly inelastic

conservation of momentum: p1i + p2i = p1f + p2f

for m1 having velocity u1 to the right, m2 initially at rest, ends with velocity v2.
x dimension: m1u1 = m1u2cosθ1 + m2v2cosθ2
y dimension:   0  = m1u2sinθ1 - m2v2sinθ2

Glancing blow: If and only if both masses are equal (like billiards), then the angle between the resulting vectors is always 90 degrees.

### Inclined plane

normal force = force perpendicular to the plane
normal force on a block resting on a slope, θ = degrees from horizontal:
f = m*g*cos(θ)
parallel force = force parallel to the inclined plane
f = m*g*sin(θ)
When parallel force > friction, it is unbalanced and objects will move down the plane
Applied force - friction = net force
static friction -
μS (mu static) = fS/N
(fS = force where static friction is overcome
N = normal force) must be overcome before the mass moves
μS = fs/N = m*g*sin(θ) / m*g*cos(θ) = sin(θ)/cos(θ) = tan(θ)
kinetic friction - normal moving friction
only one type of friction applies at a time

### Spring and Lever

Hooke's law for springs: F=-kx, k=spring constant, x = displacement

Fulcrum: t = r * f (torque = radius * force) just add the torques for multiple objects on one side of a fulcrum

### SHM - Simple Harmonic Motion

ma = -kx

angular frequency ω = √(k/m)

period of oscillation T = 2π √(m/k) (horizontal or vertical springs)

In a vertical spring, the weight Mg of the body produces an initial elongation to equilibrium, such that Mg − kyₒ = 0.

If y is the displacement from this equilibrium position the total restoring force will be Mg − k(yₒ + y) = −ky

### Projectile fired at an angle

Vx = Vo*cos(θ)
Vy = Vo*sin(θ) - gt
x = Vx*t
y = Vy*t - g*t²/2

projectile follows the shape of a parabola

y = Ax² + Bx
y = -gx²/(2(VoCos(θ))²) + xtan(θ)
time of flight: t = 2Vosin(θ)/g
max height: H = (Vosin(θ))²/2g
distance: x = sin(2*θ)*Vo² / g
Vo = initial velocity
Can use 2sin(θ)cos(θ) = sin(2θ)
if filling in t with time of flight in the x = Vx*t formula

Vf² = Vi² + 2ad ?

### Buoyant force

pressure P = F/A (force/area)

hydrostatic gauge pressure: P = pgh, p = density of fluid, g=gravity, h=height (depth)

buoyant force Fb = Fup - Fdown

Fb = pgVf,  where Vf = volume of displaced fluid, and density * volume = mass, so
Fb = mf*g,  where mf = mass of displaced fluid
=> buoyant force depends on mass of displaced fluid, not the mass of the object

### Gravity

gravitational constant between two bodies

F = G * m1 * m2 / r²
and g = G * m1 / r²
gE (gravity Earth)  = 9.8 m/s²

### Kinematics

no use of forces in the equations

typical equations:
d = vo*t + 1/2*a*t²
d = (vo + vf)/2 * t