52708A
Let two particles of equal mass collide. Particle 1 has initial velocity , directed to the
right, and particle 2 is initially stationary.
Answer:
This type of problems have a common approach, simply the masses and the initial
velocities are different. Let us first derive the equations in common.
Let the masses of the particles be m1 and m2 and their initial velocities(along line
joining) are u1 and u2 respectively (u1 must be > u2 for collision), and after collision
moving with velocities v1 and v2 respectively(v2 >v1 ; separation)
According to law of conservation of linear momentum and of kinetic energy if the
collision is elastic.
m1u1 + m2u2 = m1v1 +m2v2 …………(1) and
(1/2)m1u12 + (1/2)m2u22 =(1/2)m1v12 + (1/2)m2v22, which gives
m1u12 + m2u22 = m1v12 + m2v22 ………(2)
To avoid square terms we can solve the equations for final velocities in this way:
rearranging the equations
m1u1 - m1v1 = m2v2 - m2u2 … ……….(1) and
m1u12 - m1v12 = m2v22 - m2u22 …….(2)
dividing 2 by 1 and simplifying we get u1 + v1 = u2 + v2
or
u1 – u2 (velocity of approach) = v2 – v1 (velocity of separation) ….(3)
using equation 1 and 3 is easier.
Part A
If the collision is elastic, what are the final velocities and of particles 1 and 2?
Give the velocity of particle 1 followed by the velocity of particle 2, separated by a
comma. Express each velocity in terms of .
1)
mv + 0 + = mv1 + mv2 gives v = v1 + v2 and
2)
v = v2 – v1
adding the two we get v2 = v
and by subtracting v1 = 0
Answer: 0, v
Part B
Now suppose that the collision is perfectly inelastic. What are the velocities and of
the two particles after the collision?
Give the velocity of particle 1 followed by the velocity of particle 2, separated by a
comma. Express the velocities in terms of .
If the collision is perfectly inelastic the particles will stick together and have same
velocity after collision v1 = v2. We cant use conservation of energy equation and hence
using first equation mv = (m + m)v1
gives v1 = v/2 so
Answer: v/2, v/2.
Part C
Now assume that the mass of particle 1 is
, while the mass of particle 2 remains
the collision is elastic, what are the final velocities and of particles 1 and 2?
. If
Give the velocity of particle 1 followed by the velocity
comma. Express the velocities in terms of .
of particle 2, separated by a
1)
2)
2mv + 0 = 2m v1 + mv2 gives 2v1 + v2 = 2v
and
v = v2 – v1
Solving the two equations we get v1 = v/3 and v2 = 4v/3
Answer: v/3, 4v/3.
Part D
Let the mass of particle 1 be and the mass of particle 2 be
. If the collision is
perfectly inelastic, what are the velocities of the two particles after the collision?
Give the velocity of particle 1 followed by the velocity of particle 2, separated by a
comma. Express the velocities in terms of .
1)
mv = mv1 +3mv2 and as perfectly inelastic collision v1 = v2: we get
v1 = v2 = v/4
Answer: v/4, v/4