Collisions REVIEW. These Problems are DUE WEDNESDAY NOVEMBE 19 from EVERYONE
ELASTIC COLLISIONS REVIEW
In an elastic collision, both momentum and kinetic energy are conserved.
Since KE is conserved, the relative velocity between the two objects remains the same.
Therefore, we can write a second equation for relative velocity.
We can write these conditions as:
𝒎𝟏 𝒗𝟏 +𝒎𝟐 𝒗𝟐 = 𝒎𝟏 𝒗′𝟏 + 𝒎𝟐 𝒗𝟐 ′ [Conservation of momentum]
And
𝐯𝟏 − 𝐯𝟐 = −(𝐯𝟏′ − 𝐯𝟐′ ) [Relative velocity]
This equation in combination with the conservation of momentum will be used to solve problems
dealing with perfectly elastic, head-on collisions. According to the relative velocity equation, the
relative velocity of the two objects before the collision equals the negative of the relative
velocity of the two objects after the collision.
EXAMPLE:
Two billiard balls move toward one another. The balls have identical masses and assume that the
collision is elastic. If the initial velocities of the balls are +30 cm/s and -20 cm/s, what is the
velocity of each ball after the collision?
SOLUTION:
(1) Start with the conservation of momentum equation. Since the masses are the same, you
can cancel the masses out.
30 cm/s + (-20 cm/s) = 𝑣1 ′ + 𝑣2 ′
10 cm/s = 𝒗𝟏 ′ + 𝒗𝟐 ′ (equation 1)
(2) Because KE is also conserved, apply the relative velocity equation, which gives:
30cm/s – (-20 cm/s) = v2 ′- v1 ′
50 cm/s = 𝒗𝟐 ′- 𝒗𝟏 ′ (equation 2)
(3) Solve the equation simultaneously and you get
𝑣1′ = −20 𝑐𝑚/𝑠 And 𝑣2′ = +30 𝑐𝑚/𝑠
Collisions REVIEW. These Problems are DUE WEDNESDAY NOVEMBE 19 from EVERYONE
That is they have exchanged velocities! This is always the case when two objects of EQUAL
mass collide elastically head on.
NOW it is your turn:
TRIAL PROBLEM:
Find the final velocity of the two balls if the ball with the initial velocity -20 cm/s has a mass
𝑐𝑚
equal to half the ball with the initial velocity of +30 cm/s. [ANSWER: 𝑣1′ = −3.0 𝑠 ; 𝑣2′ =
+47 𝑐𝑚/𝑠]
PRACTICE PROBLEMS
Elastic Collisions
(1) A5.00 g object moving to the right at 20.0 cm/s makes an elastic head-on collision with a
10.0 g object that is initially at rest. Find (a) the velocity of each object after the collision
and (b) the fraction of the initial KE transferred to the 10.0 g object. [(a) 𝑣1′ =
−6.67
𝑐𝑚
𝑠
𝑎𝑛𝑑 𝑣2′ =13.3 cm/s (b) 0.88]
(2) A 10.0 g object moving to the right at 20.0 cm /s makes an elastic head-on collision with
a 15.0 g object moving in the opposite direction at 30.0 cm/s. Find the velocity of each
𝑐𝑚
object after the collision. [𝑣1′ = −40.0 𝑠 ; 𝑣2′ = 10.0𝑐𝑚/𝑠]
(3) A 25.0 g object moving to the right at 20.0 cm/s overtakes and collides elastically with a
10.0 g object moving in the same direction at 15.0 cm/s. Find the velocity of each object
𝑐𝑚
after the collision. [𝑣1′ = 17.1 𝑠 ; 𝑣2′ = 22.1𝑐𝑚/𝑠]
(4) A billiard ball rolling across a table at 1.50 m/s makes a head-on elastic collision with an
identical ball. Find the speed of each ball after the collision:
𝑚
a. When the second ball is at rest. [𝑣1′ = 0.0 𝑠 ; 𝑣2′ = 1.50𝑚/𝑠]
b. When the second ball is moving toward the first at a speed of 1.00 m/s. [𝑣1′ =
−1.00
𝑚
𝑠
; 𝑣2′ = 1.50 𝑚/𝑠]
c. When the second ball is moving away from the first at a speed of 1.00 m/s.
𝑚
[𝑣1′ = 1.00 𝑠 ; 𝑣2′ = 1.50 𝑚/𝑠]
Collisions REVIEW. These Problems are DUE WEDNESDAY NOVEMBE 19 from EVERYONE
(5) Two blocks of masses m1 = 2.00 kg and m2 = 4.00 kg are released from rest at a height of
5.00 m on a frictionless track as shown below. The blocks undergo an elastic head-on
collision. (a) Determine the velocity of each block just before the collision (b) Determine
the velocity of the blocks immediately after the collision. (c) Determine the maximum
height to which the masses will rise after the collision.
m1= 2.00 kg
m2 =4.00 kg
5.00 m
5.00 m
ANSWERS: (a) 9.90 m/s for both (b) [𝑣1′ = −16.5 𝑚𝑠 ; 𝑣2′ = 3.30 𝑚/𝑠] (c) [h1f = 13.9 m and h2f = 0.56 m]
Additional problems you need to practice:
(6) A 7.0 g bullet is fired into a 1.5 kg ballistic pendulum. The bullet emerges from the block
with a speed of 200 m/s, and the block rises to a maximum height of 12 cm. Find the
initial speed of the bullet. [v=528 m/s]
(7) An 8.00 g bullet is fired into a 250 g block that is initially at rest on the edge of a table of
1.00 m height. The bullet remains in the block and after the collision the block lands 2.00
m from the table’s base. Determine the initial speed of the bullet. [ v = 142.9 m/s (about
320 mph)]
(8) A 12.0 g bullet is fired horizontally into a 100 g wooden block that is initially at rest on a
frictionless horizontal surface and connected to a spring of k = 150 N/m. If the bulletblock system compresses the spring by a maximum of 80.0 cm, what was the velocity of
the bullet at impact with the block? [273 m/s].
(9) Two carts of equal mass, m = 0.250 kg, are placed on a frictionless track that has a light
spring of force constant k = 50.0 N/m attached to one end of it. The first car is given an
initial velocity of 3.00 m/s to the right and the second cart is initially at rest. If the cars
collide elastically, find (a) the velocity of the cars just after the first collision [0.0 m/s and
3.0 m/s] (b) the maximum compression in the spring [ x = 0.21 m]
3.00m/s
v=0
k=50.0 N/m