Linear momentum collision problem calculations 2

Questions 

1.     A toy truck with a mass of 3 kg was moving at 3 ms^-1 and hit with another toy truck of 1 kg and was moving with a velocity of 1 ms^-1, in the opposite direction. After collision, both trucks moved together in the same direction, calculate the common velocity of the two objects after collision.

 

m₁u₁ + m₂u_{2 =  (}m₁+ m₂₎v

            (3)(3) + (1)(-1) = (3+1)v

            9 – 1 = 4v

            v = 2 ms^-1

 

2.     A hunter shot a 100g bullet from a 1.5kg gun. If the bullet travelled 200 ms-1 after being triggered, what is the backward jerk of the gun. (Think final backward velocity of the gun).

 

Total initial momentum = Total final momentum = 0

m₁u₁ + m₂u_{2 =} 0

1.5(v) + 0.1(200) = 0 ß make sure to convert 100g to kg

1.5v +20 = 0

v = - 13.3 ms-1   ß Value is negative because gun moved in opposite direction

 

3.     A trolley of mass 2 kg was in a stationary condition, before a 5 g sticky plasticine was thrown into it with a velocity of 500 ms-1. After hitting the trolley, the plasticine sticks into it. Calculate the final velocity of both the trolley and plasticine?

Use the classic formula of momentum conservation

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

0.005 (500) + 2 (0) = (0.005+ 2)v

2.5 = 2.005v

 v = 1.245 ms^-1        

 

Linear momentum calculation 1 (Collisions)

The principle of conservation of momentum:

Total momentum before collision = Total momentum after collision


Inelastic collision 

1. An object of mass = 2 kg with initial velocity of  30 m/s hit a stationary object of mass = 4 kg. After the collision, both objects move together with identical velocity. Calculate the final velocity of both objects.

Total momentum before collision 

= m₁u₁ + m₂u₂

_{= 2(30) + 4 (0)}

= 60 kg ms^-1

Total momentum after collision = 60 kg ms^-1

= m₁v₁ + m₂v₂

= 2v + 4v

= 6v

Remember

60 kg ms^-1 = 6 v

Final velocity for both objects, v = 10 ms^-1 

Elastic collision 

1. An object of mass (object 1) = 2 kg with initial velocity of  40 m/s hit an object of mass (object 2) =3 kg with initial velocity = 20 m/s. After the collision, object 1 moves with v₁ = 30 m/s. Calculate the final velocity for object 2.

Total momentum before collision

= m₁u₁ + m₂u₂

= 2 x 40 + 4 x 20

= 160 kg ms^-1

Total momentum after collision

= m₁v₁ + m₂v₂

= 2 x 30 + 4 v₂

= 60 + 4 v₂

Final velocity of object 2

160 = 60 + 4 v₂

4 v₂ = 100

v₂ = 25 ms^-1

3. Object 1 (m₁ = 2 kg) moves with  initial velocity (u₁= 20 m/s) and hits object 2 (m2 = 1 kg) (u2 = - 5 m/s) which moves in the opposite direction. After the collision, Object 1 moves with velocity = 15 m/s. Calculate the velocity for object 2, assuming that the collision is elastic.

Total momentum before collision

= 2 x 20 + 1 x (-5) negative denotes opposite direction

= 35 kg ms^-1

Total momentum after collision

= 2 x 15 + 1 x v₂

= 30 + v₂

Final velocity of object 2, moving in the same direction

35 = 30 + v₂

v_{2 =}  5 m/s

Solving problems, questions, calculations, linear motion 2

Look at the diagram above. It is a displacement-time graph for a moving object.

Questions:

a) What is the displacement of the object at t = 2 s?

Answer = 20 m

b) When does the object cease to move?

Answer = From t = 4 s until t = 8 s

c) When does the object starts to move in the opposite direction?

Answer = At t = 8 s.

d) What is the velocity of the object at time

i) t = 6 s?

Answer = the object is not moving at all, hence velocity is zero.

ii) t = 9 s?

Gradient of the graph, as t = 9 is in the middle of the movement.

Hence v = - 40 / 2 = - 20 m/s

* Note the negative sign which indicates that the object is moving in the opposite direction.

e) Calculate the total distance traveled?

40m + 80m = 120 m, note that the direction of the movement does not matter here as distance is a scalar quantity.

f) Calculate the total displacement traveled?

+40m = ( - 80m) = -40m

Note that the total displacement is negative indicating the final position is -40m opposite the first direction of travel.

g) What is the average speed of the object?

Average speed =  distance / time
= 120 / 12 = 10 m/s

h) What is the average velocity of the object

Average velocity  = total displacement / time

= -40 / 12
= 3.33 m/s