Transcript Collisions in One Dimension
Elastic Collision of Two Bodies in One Dimension: The Generalized Case Paul Robinson
Initial Conditions Block 1, of mass m 1 , moves across a frictionless surface with speed v 1i . It collides elastically with block 2, of mass m 2 , which is at rest. After the collision, block 1 moves with speed v 1f , while block 2 moves with speed v 2f . What are v 1f and v 2f ?
Part 1: Isolate v 2f Using the conservation of momentum, we isolate v 2f variables.
in terms of the other
m v
1 1
i m v
2 2
f
m v
1 1
f
m v
1 1
f
m v
2 2
f
m v
1 1
i v
2
f
m
1
m
2
v
1
f
v
1
i
Part 2: Solve for v 1f Take our v into the conservation of kinetic energy.
2f expression, and plug it
m v
1 1
i
2
m v
1 1
f
2
m v
2 2
f
2
m v
1 1
i
2
m v
1 1
f
2
m
2
m
1
m
2 2 2
v
1
f
v
1
i
2
Part 2 cont… Now, take the updated KE equation and solve for v group the v 1 2 1f in terms of the given constants. First we cancel out, then on one side, and then factor and cancel out again.
m v
1 1
i
2
m v
1 1
f
2
m
1 2
m
2 2
v
1
f
v
1
i
2
m
2
m v
1 1
i
2
m v
1 1
f
2
m
1 2
m
2 2
v
1
f
v
1
i m
1
m
1
m
2
m
2
v
1
f
v
1
i
2
m
2
m
1
v
1
i
2
v
1
f
2
m
1 2
m
2
v
1
i
v
1
f
v
1
i
v
1
f
v
1
f m
1
m
2
v
1
i
v
1
f
2
v
1
i
v
1
i
v
1
f
m
1
m
2
v
1
f
v
1
i
2
Solve for v 2f Take the v 1f expression and plug into the conservation of momentum equation. Then, simply solve for v 2f
m v
1 1
i
m v
1 1
f
m v
2 2
f m v
1 1
i m v
2 2
f f
m
1
m m
1 1
m
2
m
2
v
1
i
m v
2 2
f
m v i
m
1
m m
1 1
m
2
m
2
v
1
i
m
1
m
1
m
1
m
1
m
2
m
2
v
1
i
v
2
f
m
1 2
m
1
m
2
v
1
i