1 Equal mass Suppose that two objects of equal mass collide. One of them is init

Question

1 Equal mass

Suppose that two objects of equal mass collide. One of them is initially stationary and

the other has velocity ~Vi. They bounce off of each other with velocities ~v1f and ~v2f. The

velocities after the collision satisfy two constraints:

a. The combined kinetic energy of the objects after the collision is the same as the kinetic

energy of the rst one before the collision.

b. The combined momentum of the objects after the collision is the same as the momen-

tum of the rst object before the collision.

Prove that ~V1f (dot) ~V2f= 0

Hint: v^2=v.v

2 Pool balls

Say that two pool balls collide elastically. After the collision there are 4 unknown variables:

The two components of each ball's velocity. Momentum conservation provides 2 equations

(two components), and energy conservation provides one more. What's the fourth equation?

Hint: Think of how they make contact with each other.

3 Center of mass frame

Suppose that two objects with masses m1 and m2 collide with velocities ~v1i and ~v2i. They

bounce off with velocities ~v1f and ~v2f. Suppose that, before the collision, the center-of-

mass of the system is stationary, i.e. ~Vcm = 0.

Prove that ~v1f = (negative)~v1i and ~v2f = (negative)~v2i

PLease Ellaborate as much as possible!
~V=vector V

Explanation / Answer

I beleive there is nothing more to explain in part 1 & 2

so I am only explaining part 3

(m1+m2) * Vcm = m1 v1i + m2 v2i

given Vcm = 0

so,

m1v1i + m2v2i = 0

v2i = - ( m1/m2) v1i

now as there is no net external firce acting on system ( during collision only internal forces act ) //

so, the velocity of center of mass of system wont change

so,

m1v1f + m2v2f = (m1+m2) Vcm = 0

so, v2f = - ( m1/m2 ) v1f

initial energy of fystem = 0.5 * ( m1v1i^2 + m2 * ( m1^2 / m2^2 ) v1i^2 )

= 0.5* (m1/m2) v1i^2 [ m1 + m2 ]

final energy = 0.5 * ( m1/m2) v1f^2 [ m1 + m2 ]

for elastic collision ,

intial energy = final

so , on equating we get .... v1i^2 = vif^2

so, v1i = + v1f or - v1f

now coefficient of restitution equation for elastic collision..

( v2f - v1f ) / ( v2i - v1i ) = -1

so, ( - ( m1 / m2 ) * v1f - v1f ) / ( -( m1/m2)*v1i - v1i ) = -1

so, v1f / v1i = -1

so, v1f = - v1i

v2f = -(m1/m2) v1f = - v2i

hence proved

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