DERIVATION # 2 FOR ELASTIC COLLISIONS - Stuyvesant High School

[Pages:2]STUDENT NAME...............................................PERIOD.............DATE.............. SPS21; HONORS PHYSICS Dr. Majewski

DERIVATION # 2 FOR ELASTIC COLLISIONS

1. Consider a one-dimensional, head-on elastic collision. One object has a mass m1 and an initial

velocity v1; the other has a mass m2 and an initial velocity v2. Use momentum conservation and

energy conservation to derive the final velocities after an elastic collision.

Picture the Problem: The mass m1 has an initial velocity v1 and collides with m2 that has initial velocity v2 .

Strategy: Combine the equations of momentum conservation and energy conservation to find the final speeds of the two masses.

Solution: 1. Set pi = pf : 2. Set Ki = Kf :

m1v1 + m2v2 = m1v1f + m2v2f

1 2

m1v12

+

1 2

m2v22

=

1 2

m1v12f

+

1 2

m2v22f

3. Rearrange the equation from step 1:

m1 (v1 - v1f ) = 1 m2 (v2f - v2 )

4. Rearrange the equation from step 2:

( ) ( )( ) m1 v12 - v12f = 1 = m1 v1 - v1f v1 + v1f

( ) m2 v22f - v22

m2 (v2f - v2 )(v2f + v2 )

5. Set the expressions in steps 3 and 4 equal to each other:

m1 (v1 - v1f ) = m1 (v1 - v1f ) (v1 + v1f ) m2 (v2f - v2 ) m2 (v2f - v2 ) (v2f + v2 )

v2f + v2 = v1 + v1f v2f = v1 + v1f - v2

6. Substitute the expression for v2f in step 5

( ) m1v1 + m2v2 = m1v1f + m2 v1 + v1f - v2

into the equation from step 1:

(m1 - m2 ) v1 + 2m2v2 = (m1 ) + m2 v1f

v1f

=

m1 m1

- +

m2 m2

v1

+

2m2 m1 + m2

v2

7. Now substitute the expression for v1f into the last equation of step 5 to find v2f :

v2f

=

v1

+

m1 m1

- +

m2 m2

v1

+

2m2 m1 + m2

v2 - v2

=

m1 m1

- m2 + m2

+1 v1

+

2m2 m1 + m2

-1 v2

v2f

=

2m1 m1 + m2

v1

+

m2 m1

- m1 + m2

v2

Insight: These complex formulae predict the outcome of any one-dimensional elastic collision. The equations get even more complex if you include two dimensions!

2. Two objects with masses m1 and m2 and initial velocities v1,i and v2,i move along a straight line and collide elastically. Assuming that the objects move along the same straight line after the collision, show that their relative velocities are unchanged; that is, show that v1,i - v2,i = v2,f - v1,f . (You can use the results given in Problem 1)

Picture the Problem: The mass m1 has an initial velocity v1 and collides with m2 that has initial velocity v2 . Strategy: Subtract the expressions for v2f and v1f found in problem 74 to prove the indicated relation.

Solution: Use the results of problem 1 to write an expression for v2f - v1f :

v2f

- v1f

=

2m1 m1 + m2

-

m1 m1

- m2 + m2

v1i

+

m2 m1

- +

m1 m2

-

2m2 m1 + m2

v2i

=

m1 m1

+ +

m2 m2

v1i

+

-m1 - m2 m1 + m2

v2i

v2f - v1f = v1i - v2i

Insight: This shows that for a head-on, elastic collision the difference in speeds remains constant. If m2 were at rest initially, then after the collision its speed relative to m1 will be v1i . If the two masses are equal, and m2 is initially at rest, then m1 stops completely and m2 leaves the collision with speed v1i .

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