Continuous Mass Flow Rockets 8.01 W08D1

Juno Lift-Off

Today’s Reading Assignment: W08D1 Young and Freedman: 8.6 Class Notes: Continuous Mass Flow

Worked Example: Stream Bouncing off Wall A stream of particles of mass m and separation d hits a perpendicular surface with speed v. The stream rebounds along the original line of motion with the same speed. The mass per unit length of the incident stream is = m/d. What is the magnitude of the force on the surface?

Category 1: Adding Mass But Not Momentum There is a transfer of material into the object but no transfer of momentum in the direction of motion of the object. Consider for example rain falling vertically downward into a moving cart. A small amount of rain has no component of momentum in the direction of motion of the cart.

Concept Question Suppose rain falls vertically into an open cart rolling along a straight horizontal track with negligible friction. As a result of the accumulating water, the speed of the cart 1.increases. 2.does not change. 3.decreases. 4.not sure. 5.not enough information is given to decide.

Fire Plane Scooping up Water

Category 2: Losing Mass But Not Momentum The material continually leaves the object but it does not transport any momentum away from the object in the direction of motion of the object. For example, consider an ice skater gliding on ice holding a bag of sand that is leaking straight down with respect to the moving skater.

Category 3: Impulse The material continually hits the object providing an impulse resulting in a transfer of momentum to the object in the direction of motion. For example, suppose a fire hose is used to put out a fire on a boat. The incoming water continually hits the boat impulsing it forward.

Category 4: Recoil The material continually is ejected from the object, resulting in a recoil of the object. For example when fuel is ejected from the back of a rocket, the rocket recoils forward.

Rocket Equation

Worked Example: Rocket Motion A rocket at time t is moving with speed v r,0 in the positive x-direction in empty space. The rocket burns the fuel at a rate dm f,out /dt =b > 0. The fuel is ejected backward with speed u relative to the rocket. a) What is the relationship between the time rate of change of exhaust mass dm f /dt, and the time rate of change of rocket mass dm r /dt? b) Find an equation for the rate of change of the speed of the rocket in terms m r (t),u, and dm r /dt and solve for v. c) Find the differential equation describing the motion of the rocket if it is in a constant gravitational field of magnitude g

Strategy : Rocket Motion Goal: Determine velocity of rocket as function of time as mass is continuously ejected at rate dm f /dt with speed u relative to rocket. System: consider all elements that undergo momentum change: rocket and fuel Using Momentum flow diagram, apply to find differential equation that describes motion.

Rocket Motion: A rocket at time t = 0 is moving with speed v r,0 in the positive x-direction in empty space. The rocket burns the fuel at a rate dm f /dt = b >0. The fuel is ejected backward with speed u relative to the rocket. The goal is to find an equation for the rate of change of the speed of the rocket in terms m r (t),u, and dm r /dt and solve for v.

State at time t 1.Rocket with total mass m r (t) moves with speed v r (t) in positive x- direction according to observer 2. Total mass consists of mass of rocket m r,0 and fuel m f (t) 3. Fuel element with mass Δm f, moves with speed of rocket v r (t) at time t, is ejected during interval [t, t+Δt] 4. x-component of momentum at time t

State at t + Δt Rocket is propelled forward by ejected fuel with new rocket speed Fuel is ejected backward with speed u relative to rocket. Relative to observer’s frame, ejected fuel element has speed x-component of system’s momentum at time t+Δt

Rocket Equation Are there any external forces at time t? Two cases: (1)Taking off (2) Negligible gravitational field Apply Momentum Principle:

Rocket Equation: Conservation of Mass and Thrust Conservation of mass: Rate of decrease of mass of rocket equals rate of ejection of mass Rocket equation: Fuel Ejection Thrust force:

Solution: Rocket Motion in Gravitational Field Rocket Equation: Fuel ejection term can be interpreted as thrust force Relative fuel ejection velocity External force Rocket equation Integrate with respect to time Solution:

Concept Question: Rocket Fuel Burn Time When a rocket accelerates in a constant gravitational field, will it reach a greater final velocity if the fuel burn time is 1.as fast as possible? 2.as slow as possible? 3.The final speed is independent of the fuel burn time? 4.I’m not sure.

Answer: Rocket Equation in Gravitational Field Fuel ejection term can be interpreted as thrust force Solution: shorter the burn time, the greater the velocity

Concept Question: Rocket with Constant Thrust If a rocket in gravity-free outer space has a constant thrust at all times while burning fuel, is its acceleration 1. constant? 2. increasing? 3. decreasing?

Table Problem: Rocket Sled A rocket sled ejects gas backwards at a speed u relative to the rocket sled. The mass of the fuel in the rocket sled is equal to one half the initial total mass m r,0 (including fuel) of the sled. The rocket sled starts from rest on a frictionless track. You may ignore air resistance. a) Derive a relation between the differential of the speed of the rocket sled, dv r, and the differential of the total mass of the rocket, dm r b) Integrate the above relation to find the speed of the rocket sled as a function of mass, v r (m), as the rocket sled speeds up. c)What is the final speed of the rocket sled after all the fuel has been burned? Express your answers in terms of the quantities u, and m r,0 as needed. d)After reaching its final speed, the sled enters a rough portion of the track that begins at x = 0 with a coefficient of kinetic friction that varies with distance μ k (x)=bx where b is a positive constant. How far D did the sled slide before it came to rest in that portion of the track? Express your answers in terms of the quantities u, b, g, and m r,0 as needed.

Next Class W08D2 Test Two Review Circular Motion, Energy, Momentum, and Collisions

Next Reading Assignment: W09D1 Young and Freedman: 1.10 (Vector Products) , 10.5