# Collisions – problems and solutions

1. Object A (3 kg) moves at a speed of 8 m/s and object B (5 kg) moves at a speed of 4 m/s. If the collision between the object A and B is perfectly elastic, what is the velocity of object A and B after the collision?

__Known :__

Mass of object A (m_{1}) = 3 kg

Mass of object B (m_{2}) = 5 kg

The speed of object A (v_{1}) = 8 ms^{–1 }

The speed of object B (v_{2}) = 4 ms^{–1}

__Wanted:__ v_{1}‘ and v_{2}‘

__Solution :__

If both objects have the different mass and the velocity of both objects after the collision is not known yet, then the velocity after collision calculated using this equation :

The direction of both objects is the same so that the velocity of both objects have the same sign. If both objects move in opposite direction then one velocity has plus sign and another velocity has minus sign.

The velocity of object A (v_{A}) after the collision :

The velocity of object B (v_{B}) after the collision :

Velocity of object A (v_{1}‘) after collision is 3 m/s and velocity of object B (v_{2}‘) after collision is 7 m/s.

2. A ball has a momentum of P, collide a wall and reflected. The collision is perfectly elastic and its direction perpendicular to the wall. What is the change of the ball’s momentum?

__Known :__

Mass of ball = m

The velocity of the ball before collision = v

Velocity of ball after collision = -v (ball reflected leftward)

Ball’s momentum before collision (p_{o}) = m v

Ball’s momentum after collision (p_{t}) = m (-v) = – m v

__Wanted:__ The change of the ball’s momentum

__Solution :__

The change of momentum :

Δp = p_{t} – p_{o}

Δp = – m v – m v

Δp = – 2 m v

Δp = -2p

Minus sign indicates the direction of the ball.

3. Two objects, A and B, have the same mass move in an opposite direction and collide with each other. A moves east a speed of V and B move west at a speed of 2V. If the collision is perfectly elastic, then what is the velocity of both objects after the collision.

__Known :__

Both objects have the same mass.

A moves east at speed of V

B moves west at speed of 2V

__Wanted:__ The velocity of both objects after the collision

__Solution :__

If both objects have the same mass and the collision is perfectly elastic, then after collision both objects change their velocity.

After the collision, A moves west at 2V and B moves east at V.

4. Two objects with the same mass move along a straight line in opposite direction as shown in the figure below. If the velocity of object 2 after the collision (v_{2}‘) is 5 m/s rightward, then what is the magnitude of the velocity of the object 1 after the collision.

__Known :__

Mass of each object = m

Velocity of object 1 before collision (v_{1}) = 8 m/s

Velocity of object 2 after collision (v_{2}) = 10 m/s

Velocity of object 2 after collision (v_{2}‘) = 5 m/s

__Wanted :__ Velocity of object 1 after collision (v_{1}‘)

__Solution :__

The collision is partially elastic.

m_{1} v_{1}+ m_{2} v_{2} = m_{1} v_{1}’ + m_{2} v_{2}’

m (v_{1} + v_{2}) = m (v_{1}’ + v_{2}’)

v_{1} + v_{2} = v_{1}’ + v_{2}’

8 + 10 = v_{1}’ + 5

18 = v_{1}’ + 5

v_{1}’ = 18-5

v_{1}’ = 13 m/s

**Inelastic collision**

5. A 10-gram bullet fired at 100 m s^{-1} , collide a block of wood at rest. Mass of block is 490 gram. The collision is inelastic. What is the speed of the block and bullet after collision.

__Known :__

mass of bullet (m_{1}) = 10 gram

Speed of bullet (v_{1}) = 100 m/s

Mass of block (m_{2}) = 490 gram

Speed of block (v_{2}) = 0 m/s (block at rest)

__Wanted :__ The speed of the block and bullet after collision

__Solution :__

m_{1 }v_{1} + m_{2} v_{2} = (m_{1} + m_{2}) v’

(10)(100) + (490)(0) = (10 + 490) v’

1000 + 0 = 500 v’

1000 =500 v’

v’ = 1000 / 500

v’ = 2 m/s

6. A truck moves at 10 m/s collide a car moves at 20 m/s. After collision, both truck and car move together at the same speed. Mass of truck is 1400 kg and mass of car is 600 kg. What is the velocity of both truck and car after collision.

__Known :__

speed of truck (v_{1}) = 10 m/s

speed of car (v_{2}) = 20 m/s

mass of truck (m_{1}) = 1400 kg

mass of car (m_{2}) = 600 kg

__Wanted :__ The velocity of both truck and car after collision

__Solution :__

m_{1} v_{1} + m_{2} v_{2} = (m_{1 }+ m_{2}) v

(1400)(10) + (600)(20) = (1400 + 600) v

14000 + 12000 = 2000 v

26000 = 2000 v

v = 13 m/s

7. A 20-gram bullet moves at 10 m/s in a horizontal direction, collide a 60-gram block rest on a floor. After collision, bullet and block move together with the same speed and the same direction. What is the speed of both bullet and block.

__Known :__

Mass of bullet (m_{P}) = 20 gram = 0.02 kg

Mass of block (m_{B}) = 60 gram = 0.06 kg

The initial speed of bullet (v_{P}) = 10 m/s

The initial speed of block (v_{B}) = 0

__Wanted :__ The speed of bullet and block after collision (v’)

__Solution :__

The collision is inelastic.

m_{P} v_{P }+ m_{B }v_{B} = (m_{P} + m_{B}) v’

(0.02)(10) + (0.06)(0) = (0.02 + 0.06) v’

0.2 + 0 = 0.08 v’

0.2 = 0.08 v’

v’ = 0.2 / 0.08

v’ = 2.5 m/s

8. Two objects, A and B, have the same mass = 1.5 kg approach each other and collide. Speed of object A (v_{A}) = 4 m/s and speed of object B (v_{B}) = 5 m/s. If the collision is inelastic, what is the speed of both objects after collision.

__Known :__

Mass of object A (m_{A}) = 1.5 kg

Mass of object B (m_{B}) = 1.5 kg

Speed of object A before collision (v_{A}) = 4 m/s (positive rightward)

Speed of object B before collision (v_{B}) = -5 m/s (negative leftward)

__Wanted :__ The speed of both objects after collision

__Solution :__

m_{A} v_{A} + m_{B} v_{B }= (m_{A} + m_{B}) v’

(1.5)(4) + (1.5)(-5) = (1.5 + 1.5) v’

6 – 7.5 = (3) v’

-1.5 = (3) v’

v’ = -1.5 / 3

v’ = -0.5 m/s

Minus sign indicates that after collision, both objects move leftward, same direction as object B.