Collisions in one dimension that conserve energy and momentum

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Collisions in one dimension that conserve energy and momentum

Spring 2014

Instructor: Steven Detweiler

Assume that we have two masses M

1 and M

2 moving with initial speeds v

1 i and v

2 i in the same direction—if the objects are actually moving towards each other then just assume that v

2

, for example, is negative.

The conservation of momentum implies that

M

1 v

1 i

+ M

2 v

2 i

= M

1 v

1 f

+ M

2 v

2 f where v

1 f and v

2 f are the speeds of the masses after the collision.

The conservation of kinetic energy implies that

1

2

M

1 v

2

1 i

+

1

2

M

2 v

2

2 i

=

1

2

M

1 v

2

1 f

+

1

2

M

2 v

2

2 f

.

First we change these equations by putting the all of the M

1 terms on the left-hand side and the M

2 terms on the right-hand side to obtain

M

1

( v

1 i

M

1

( v

2

1 i

− v

1 f

) = M

2

( v

2 f

− v

2

1 i

) = M

2

( v

2

2 f

− v

1 f

)

− v

2

2 i

) .

(1)

(2)

Now, we divide the first of these equations by the second so that the masses cancel each other out, and we have v 2

1 i

− v 2

1 f v

1 i

− v

1 f

= v 2

2 i v

2 i

− v 2

2 f

− v

2 f

.

But, we know that a 2 − b 2 = ( a − b )( a + b ), so our equation above is equivalent to

(3) v

1 i

+ v

1 f

= v

2 f

+ v

2 i

.

(4)

At this point the algebra gets a bit challenging. But, we choose any two of the three equations (1), (2) or (4) to solve for the two unknowns,

For example first use v

1 f and v

2 f

.

v

2 f

= v

1 i

+ v

1 f

− v

2 i

, (5) from equation (4), and use this equation to substitute for v

2 f solve for v

1 f into equation (1), and then v

1 f

=

M

1

M

1

− M

2

+ M

2 v

1 i

+

M

1

2 M

2

+ M

2 v

2 i

.

By switching the subscripts 1 and 2, this becomes

(6) v

2 f

=

M

2

M

1

− M

1

+ M

2 v

2 i

+

M

1

2 M

1

+ M

2 v

1 i

.

(7)

All of the above equations, become much easier to interpret, if the situation that you have has one of the masses initially at rest.

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