-Motion of the Center of Mass-Rocket Propulsion

AP Physics CMrs. Coyle

Motion of the Center of Mass

The center of mass of a system of particlesmoves as if the total mass of the particleswere concentrated there and acted uponby 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 VCM= mivi

M

Acceleration of CM aCM= miai

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: vB = -2.857m/s vCM = 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

CM toti i ii i

M m v v p p

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)

CM ii

M Fa

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: vA =vB =42.4m/svCM before and after =30m/s g=-10m/s2

Rocket PropulsionBefore After

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: vInitial momentum: pi = (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-ve )

M v = vem =-veM then integrate

ve : velocity of ejected fuel relative to the rocket

Equation for v for Rocket Propulsion

• The higher the ve , the higher the v .

• The higher the Mi/Mf , the higher the v.

ln if i e

f

Mv v v

M

Thrust force exerted on the rocket by the exhaust gases:

FThrust =

• The higher the ve , the higher the thrust.

• The higher the burn rate (dM/dt), the higher the thrust.

e

dv dMM vdt dt

Example #50A size C5 model rocket engine has an averagethrust of 5.26N, a fuel mass of 12.7g and an initialmass 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