In these questions you will use the \"center of mass frame\", that is, a referen
ID: 1433173 • Letter: I
Question
In these questions you will use the "center of mass frame", that is, a reference frame moving at the velocity of the center of mass of the colliding objects, to solve the problem. You must understand that momentum of the center of mass remains unchanged in the collision.
Two spheres of equal radius and masses 5.00 kg and 20.0 kg are moving towards each other along the same straight line along across a frictionless, horizontal surface as illustrated below. Vi1 = +10.0m/sandVi2 = 5.00m/s respectively. During the collision, 25% of the total initial kinetic energy is lost.
(i) Find the velocity of the center of mass of this system.
(ii) What are the velocities of each sphere in the center of mass reference frame (i.e., from the point of view of someone moving with the velocity of the center of mass)?
(iii) What is total initial momentum in the center of mass reference frame? (You should find that this is zero!)
(iv) What are the initial and final total KEs in the center of mass frame?
(v) Find the final velocities of the two objects in the center of mass reference frame.
(vi) Now find the final velocities of the two objects in the laboratory frame.
Explanation / Answer
m1 = 5 Kg
m2 = 20 Kg
vi1 = 10 m/s
vi2 = -5 m/s
a)
velocity of centre of mass = (m1 * vi1 + m2 * vi2)/(m1 + m2)
velocity of centre of mass = (5 * 10 - 20 * 5)/(5 + 20)
velocity of centre of mass = -2 m/s
the velocity of centre of mass is -2 m/s
b)
from the centre of mass frame
velocity of ball 1 = 10 - (-2) m/s
velocity of ball 1 = 12 m/s
velocity of ball 2 = -5 - (-2) m/s
velocity of ball 2 = -3 m/s
the velocity of ball 2 is -3 m/s
iii)
TOtal initial momentum in centre of mass frame = 12 * 5 - 3 * 20
TOtal initial momentum in centre of mass frame = 60 - 60
TOtal initial momentum in centre of mass frame = 0 Kg.m/s
iV)
in the centre of mass frame
initial total kinetic energy = 0.5 * 5 * 12^2 + 0.5 * 20 * 3^2
initial total kinetic energy = 450 J
the initial total kinetic energy is 450 J
final total kinetic energy = (1 - 0.25) * total initial kinetic energy
final total kinetic energy = 0.75 * 450 J
final total kinetic energy = 337.5 J
the final total kinetic energy is 337.5 J
v)
let the final velocities is v1f and v2f
Using conservation of momentum
5 * v1f + 20 * v2f = 0 ---(1)
for kinetic energy
0.5 * 5 * v1f^2 + 0.5 * 20 * v2f^2 = 337.5 ---(2)
solving 1 and 2
v1f = - 10.3 m/s
v2f = 2.6 m/s
the final velocity of ball 1 is -10.3 m/s
final velocity of ball 2 is 2.6 m/s