Equations of Motion
What is an equation of motion?
An equation that describes the behavior of a system in terms of its motion as a function of time.
Give a formula for linear drag and quadratic drag.
Flin=-bv
Fquad=-cv2
How can rockets accelerate upwards without acted on by an outside force if momentum is conserved?
If the system is acted upon by an outside force.
Given the two vectors b = x + y and c = x + z (pretend x, y, and z have hats), find b+c, 5b+2c, b⋅c, and b×c. (Taylor 1.1)
b+c=2x+y+z
5b+2c=7x+5y+2z
b⋅c=1
b×c=x-y-z
Give an example of an equation of motion.
[various answers]
Explain what causes objects to have a "terminal velocity".
Drag increases with velocity. The terminal velocity is the velocity at which the force of drag balances the force of gravity, and there is no net force on the object.
Give a formula relating the acceleration of a rocket to the exhaust velocity, exhaust force, and change in mass.
m(dv/dt) = -(dm/dt)vext + Fext
Give a formula for angular momentum in terms of momentum and radius.
L = r x p
(pretend these are vectors and the x is a cross product)
Write eiθ in terms of sin and cos.
eiθ = cosθ + isinθ
Find the general solution to the differential equation:
df/dt = Cf
for some unknown function f(t) and constant C.
f = Ae^Ct
In what situations does the linear drag dominate? In what situations does the quadratic drag dominate?
Linear drag dominates in the case of low speeds and viscous fluids. Quadratic drag dominates in the case of high speeds and not-very-viscous fluids.
A rocket with initial mass m0 begins accelerating in free space and burns 0.9m0 of fuel. What is its final speed (expressed as a multiple of vex)?
2.3vex
A shell traveling with speed v0 exactly horizontally and due north explodes into two equal-mass fragments. It is observed that just after the explosion one fragment is traveling vertically up with speed v0. What is the velocity of the other fragment? (Taylor 3.2)
v=(2v0, -v0, 0) = sqrt(5)v0 at angle 26.6° below the horizontal.
What is the general solution to the differential equation u''=-w2u? To the differential equation v'=kv?
u(t) = C1cos(wt) + C2sin(wt)
v(t) = Cekt
I (being big and strong) push a 50 kg rock by applying a 1000 N force. The ground has a coefficient of kinetic friction of 1.0. If the initial velocity is 0, what is the velocity of the rock after 10 s? (Let g=10m/s^2).
100 m/s
Find the terminal speeds in air of (a) a steel ball bearing of diameter 3 mm, (b) a 16-pound steel shot, and (c) a 200-pound parachutist in free fall in the fetal position. In all three cases, you can safely assume the drag force is purely quadratic. The density of steel is about 8 g/cm3 and you can treat the parachutist as a sphere of density 1 g/cm3. (Taylor 2.23). (HINT: vter=sqrt(mg/γD2), and γ=0.25kg/m3 in all three cases).
a) vter = 22 m/s
b) vter = 140 m/s
c) vter = 107 m/s
Consider a rocket (initial mass m0) accelerating from rest in free space. At first, as it speeds up, its momentum p increases, but as its mass m decreases p eventually begins to decrease. For what value of m is p maximum? (Taylor 3.10)
p maximized when ln(m0/m)=1 or m=m0/e
Bryan and I are battling in space! Our spaceships are initially stationary with respect to each other, both with an initial mass of 500,000 kg. Bryan shoots a 100,000 kg missile at my spaceship with a velocity of 2 km/s. The missile strikes my spaceship and gets lodged in the hull. What is the relative speed of our two spaceships?
833 m/s
A cheetah starts at rest and begins running, with its acceleration given by dv/dt=(1/v)e-v^2. Use separation of variables, then integrate with u-substitution, to find the cheetah's velocity as a function of time.
v = 0.5ev^2 - 0.5
A student kicks a frictionless puck with initial speed v0, so that it slides straight up a plane that is inclined at an angle θ above the horizontal. (a) Write down Newton's second law for the puck and solve to give its position as a function of time. (b) How long will the puck take to return to its starting point? (Taylor 1.37) (HINT: define coordinates such that +x is up the slope)
a) y(t)=0, z(t)=0, x(t)=v0t - 0.5gt2sinθ
b) t=2v0/(g sinθ)
A mass m has velocity v0 at time t=0 and coasts along the x axis in a medium where the drag force is F(v)=-cv3/2. Find v in terms of the time t and the other given parameters. At what time (if any) will it come to rest? (Taylor 2.8)
v(t)=v0/(1+ct sqrt(v0)/2m)2
A rocket (initial mass m0) needs to use its engines to hover stationary, just above the ground. (a) If it can afford to burn no more than a mass λm0 of its fuel, for how long can it hover? [Hint: Write down the condition that the thrust just balance the force of gravity. You can integrate the resulting equation by separating the variables t and m. Take vex to be constant.] (b) If vex = ~3000 m/s and λ = ~10%, for how long could the rocket hover just above the earth's surface? (Taylor 3.8).
a) tmax = -ln(1-λ)vex/g
b) tmax = 32 s
A uniform spherical asteroid of radius R0 is spinning with angular velocity ω0. As the aeons go by, it picks up more matter until its radius is R. Assuming that its density remains the same and that the additional matter was originally at rest relative to the asteroid (anyway on average), find the asteroid's new angular velocity. (You know from elementary physics that the moment of inertia is (2/5)MR2.) What is the final angular velocity if the radius doubles? (Taylor 3.29)
ω=ω0(R0/R)5. If R=R0/2, then ω=ω0/32.
A uniform thin sheet of metal is cut in the shape of a semicircle of radius R and lies in the xy plane with its center at the origin and diameter lying along the x axis. Find the position of the CM using polar coordinates. (Taylor 3.21)
X=Z=0, Y=4R/3π