Conversation of Momentum Formula
Consider an object A has mass ‘m’, and object B has mass ‘m2’. The initial velocity of an object A is ‘u’, and initial velocity of an object B is ‘u2’,
We know the formula of momentum,
So, Initial momentum (A) = m1u1
& Initial momentum (B) = m2u2
Suppose these two objects are collide. The force on A due to B is F1. Then final velocity will become V1.
Momentum of A after collision = m1v1 —— (1)
A also exerts an equal force on B but in opposite direction as shown in above fig. then velocity after collision is v2
Momentum of B after collision = m2v2 —— (2)
F2 = -F1 we know that force is equal to the product of mass and acceleration
m2a2 = -m1a1
∴ m2 (v2 – u2)/t = -m1 (v1 – u1)/t ——– a = (v – u)/t
∴ m2 (v2 – u2) = -m1 (v1 – u1) —– (3)
∴ m2v2 – m2u2 = -m1v1 + m1u1
∴ (m2v2 + m1v1) = (m1u1 + m2u2)
∴ (m1u1 + m2u2) = (m1v1 + m2v2) —– (4)
Equation (4) is required equation
From this equation we can say that,
The magnitude of total initial momentum = The magnitude of total final momentum
“When no external force acts on objects, their total momentum remains constant. It does not change”.
Numericals on Conversation of Momentum
(1) There are two balls of masses 2 kg and 4 kg respectively are at rest. The 2nd ball of mass 4 kg moves with velocity 3 m/s2 find the velocity of the ball at mass 2 kg with respect to ground.
Solution:
Given,
m1 = 2 kg
m2 = 4 kg
v1 =?
v2 = 3 m/s2
We know that law at conservation of momentum
P initial = 0, balls are at rest
P final = P1 + p2
P final = p1 + p2
= m1v1 + m2v2
= 2 kg. v1 + 4 kg. 3 m/s2
Pi = Pf
0 = 2.v1 + 12kg m/s2
V1 = 6 m/s
Velocity of 2nd ball is 6 m/s.
(2) There are two cars, car A and car B. Mass of car A is 1700 kg, travelling at 27 m/s collide with car B of mass 1500 kg travelling at 25 m/s in same direction. After collision the velocity of car A becomes 20 m/s. Calculate the velocity of car B after the collision?
Solution:
Given,
Mass of car A (m1) = 1700 kg, Initial velocity of car A (u1) = 27 m/s
Mass of car B (m2) = 1500 kg, initial velocity of car B (u2) = 25 m/s
Final velocity of car A (v1) = 20 m/s, final velocity of car B (v2) =?
We know the law of conservation of linear momentum
Total initial momentum = Total final momentum
m1u1 + m2v1 = m1v1 + m2v2
(1700×27) + (1500×25) = (1700×200 + (1500×v2)
∴ 45900 + 37500 = 34000 + 1500v2
∴ 83400 = 34000 + 1500v2
1500v2 = 83400 – 34000
∴ v2 = 49400/1500
∴ v2 = 32.93 m/s
The velocity of car B after collision is 32.93 m/s.