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Elastic Collisions in One Dimension
Introduction
| Elastic Collision |
Collision in which Kinetic Energy is
Conserved.
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We are going to restrict problems in this section to elastic
collisions in one dimension. Two particles are moving along the same axis
with initial velocities of v1 and v2. After the
collision, their velocities are v'1 and v'2. A
positive and negative direction must be specified.
Conservation of Momentum m1v1
+ m2v2 = m1v'1
+ m2v'2
Conservation of
Energy 
m1v12
+ m2v22 =
m1v'12
+ m2v'22
Rephrasing
the Conservation of Momentum
m1( v1 - v'1
) = m2 ( v'2 - v2 )
Rephrasing
the Conservation of Energy
m1( v12 - v'12
) = m2 ( v'22
- v22 ) => m1(
v1 - v'1 )( v1 + v'1
) = m2 ( v'2 - v2 )( v'2
+ v2 )
Remember
a2 - b2 = ( a - b ) ( a + b )
Dividing
Conservation of Energy by Conservation of Momentum
v1 + v'1 = v'2
+ v2
Rewriting
this equation
v1 - v2 =
v'2 - v'1
This is the best form to use when working problems.
Important
interpretation of this equation - For any elastic collision, the relative speed
of the two particles after the
collision is the same as the relative collision before the
collision. This is independent of the masses of the particles.
Table of Contents
Problems
| 1. |
A particle of mass m moving with at speed v has
an elastic collision in one dimension with a stationary particle of the same
mass. What are the speeds of the two particles.
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| 2. |
A proton, 1.01 amu, is shot at a stationary
carbon-12 nucleus, 12.00 amu, with a velocity of
5.00 x 10 4 m/g. What are their velocities after an elastic
collision in one dimension?
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| 3. |
A golf ball, 0.750 kg, is thrown at a golf
ball, 50.0 g. The golf ball is moving at 90.0 km/h to the right
while the golf ball is moving 15.0 m/g to the left. What are their velocities
after an elastic collision in one dimension?
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| 4. |
A car, 1800. kg, going 150. km/h rear ends a
truck, 5500. kg going 100. km/h. What are their velocities after an elastic
collision in one dimension? |
Table of Contents
Answers
| 1. |
A particle of mass m moving with at speed v has
an elastic collision in one dimension with a stationary particle of the same
mass. What are the speeds of the two particles.
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Conservation of
Momentum mv
= mv'1 + mv'2
=> v = v'1 + v'2
(Only true because of same mass)
Conservation
of
Energy
v = v'2 - v'1
(Remember, 2nd ball was initially at rest.)
We now have two unknowns and two independent equations.
We can now solve for the two unknowns.
v'1 + v'2
= v'2 - v'1
(both sides are equal to
v) solve for v'1
2v'1 = v'2
- v'2 =
0 or v'1
= 0
Now substitute into either of the original equations.
v = v'1 + v'2
= 0 + v'2 thug
v'2 = v
Pool players observe this if no spin is
applied to the ball. It is very important that the two balls have
the same mass.
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| 2. |
A proton, 1.01 amu, is shot at a stationary
carbon-12 nucleus, 12.00 amu, with a velocity of
5.00 x 10 4 m/g. What are their velocities after an elastic
collision in one dimension?
|
|
Conservation of
Momentum mpvp
= mpv'p + mcv'c
Conservation
of
Energy
vp = v'c
- v'p
Solve Conservation of Energy for v'p
v'p = v'c - vp
and Substitute into Conservation of Momentum.
mpvp = mpv'c
- mpv'p
+ mcv'c
Now solve for v'c
2 mpvp
2 x 1.01 amu x 5.00 x 10 4 m s -1
v'c = -------------- =
--------------------------------------------- = 7.76 x 10 3 m s
-1
mp
+ mc
1.01 amu + 12.00 amu
Solve for v'p using Conservation of
Energy
v'p = v'c
- vp = 7.76 x 10 3 m s -1
- 5.00 x 10 4 m s -1 = - 4.22 x 10
4 m s -1
Observe
that the proton has bounced backward after the collision.
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| 3. |
A golf ball, 0.750 kg, is thrown at a golf
ball, 50.0 g. The golf ball is moving at 90.0 km/h to the right
while the golf ball is moving 15.0 m/g to the left. What are their velocities
after an elastic collision in one dimension?
|
|
90.0 km 1
h 1000 m
------------ x ---------- x ------------ = 25.0 m g -1
= vs
Right is positive
and Left is negative.
h 3600 g
1 km
Conservation
of Momentum msvs
+ mgvg = msv's
+ mgv'g
Conservation
of
Energy
vs - vg
= v'g - v's
Solve Conservation of Energy for v'g
v'g = vs
- vg + v's
Substitute into Conservation of Momentum.
msvs
+ mgvg = msv's
+ mgvs - mgvg
+ mgv's
Now solve for v's
msvs
+ 2 mgvg - mgvs
750. g x 25.0 m/gs + 2 x 50.0 g x ( - 15.0 m/s ) - 50.0 g x 25.0 m/s
v's = ---------------------------------
=
-------------------------------------------------------------------------------------
ms + mg
750. g + 50.0 g
v's = 20.0
m/s ( To the Right )
Now Substitute into the equation for v'g.
v'g = vs
- vg + v's = 25.0 m/s
+ 15.0 m/s + 20.0 m/s = 60.0 m/s
(To the Right)
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| 4. |
A car, 1800. kg, going 150. km/h rear ends a
truck, 5500. kg going 100. km/h. What are their velocities after an elastic
collision in one dimension?
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Conservation of
Momentum mcvc
+ mtvt = mcv'c
+ mtv't
Conservation
of
Energy
vc - vt
= v't - v'c
Solve Conservation of Energy for v'g
v't = vc
- vt + v'c
Substitute into Conservation of Momentum.
mcvc
+ mtvt = mcv'c
+ mtvc - mtvt
+ mtv'c
Now solve for v'c
mcvc
+ 2 mtvt - mtvc
1800. g x 150. km/h + 2 x 5500. kg x 100. km/h - 5500 kg x 150. km/h
v'c = ---------------------------------
=
------------------------------------------------------------------------------------------
mc + mt
1800. kg + 5500. kg
v'c = 74.7
km/h ( To the Right )
Now Substitute into the equation for v'g.
v't = vc
- vt + v'c
= 150. km/h - 100. km/h + 74.7 km/h = 124.7
km/h (To the Right)
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Table of Contents |
