-Motion of the Center of Mass -Rocket Propulsion AP Physics C Mrs. Coyle

Motion of the Center of Mass The center of mass of a system of particles moves as if the total mass of the particles were concentrated there and acted upon by the resultant force.

-Parabolic Path of the Center of Mass of the Wrench -Other points on the wrench travel different paths.

Parabolic Path of Fireworks

Velocity and Acceleration of CM Velocity of CM V CM =  m i v i M Acceleration of CM a CM =  m i a i M

Example 3: Motion of CM A skater A of mass 50kg pushes off a skater B of mass 35kg. The a velocity of A is 2 m/s. What is: a)the velocity of B after the push. b)the velocity of the CM after the push. Ans: v B = m/s v CM = 0

Momentum of the System of Particles The momentum can be expressed as The total linear momentum of the system equals the total mass multiplied by the velocity of the center of mass

Forces In a System of Particles The acceleration of the CM is caused by the net force on the system. (the internal forces cancel out)

The Momentum of a System of Particles in the Absence of an External Force is Conserved.

Example 4: Motion of Firecracker CM A firecracker is launched straight up with an initial speed of 50 m/s. 2 sec later it explodes into two equal pieces each at 45 degrees from the vertical. a)What was their initial speed? b)What was the velocity of the CM right before and right after the explosion? c)What was the acceleration of the CM before and after the explosion? Ans: v A =v B =42.4m/s v CM before and after =30m/s g=-10m/s 2

Rocket Propulsion BeforeAfter

What info is needed for rocket propulsion? An expression for  v. An expression for the thrust force.

Rocket Propulsion-Before the gas is ejected Initial mass of rocket + fuel: M +  m Initial velocity of rocket: v Initial momentum: p i = (M +  m) v

Rocket Propulsion- After the gas ejection Final time: t +  t Final mass of Rocket: M Amount of fuel ejected:  m Final speed v +  v

Motion of Center of Mass of Rocket The center of mass of the system of the rocket and gas, moves independently of the propulsion process, because the propulsion force acting is internal.

From Conservation of Momentum: Momentum Before = Momentum After (M +  m) v = M(v+  v) +  m(v-v e )  M  v = v e  m =-v e  M then integrate v e : velocity of ejected fuel relative to the rocket

Equation for  v for Rocket Propulsion The higher the v e, the higher the  v. The higher the M i /M f, the higher the  v.

Thrust force exerted on the rocket by the exhaust gases: F Thrust = The higher the v e, the higher the thrust. The higher the burn rate (dM/dt), the higher the thrust.

Example #50 A size C5 model rocket engine has an average thrust of 5.26N, a fuel mass of 12.7g and an initial mass of 25.5g(including the fuel). The duration of its burn is 1.90s. A) What is the average exhaust speed of the engine? B) If the engine is placed in a rocket body of mass 53.5g, what is the final velocity of the rocket if it is fired in outer space? Assume the fuel burns at a constant rate. Ans: a)787m/s, b) 138m/s