A particle (A) of mass M1 travelling with velocity of V1 collide perfectly with a particle (B) of mass M2 travelling with velocity V2 (less than V1) show that the velocity V1 of (A) after the impact is given by,
V1=(M1- M2/M1+M2)U1+2M2 V2/M1+M2
To show that the velocity (V1) of particle A after the impact is given by:
V1 = [(M1 - M2)/(M1 + M2)]V1 + [2M2/(M1 + M2)]V2
We can use the principle of conservation of linear momentum, which states that the total momentum of an isolated system remains constant before and after the collision.
Before the collision:
The momentum of particle A is given by the mass (M1) of particle A multiplied by its initial velocity (V1):
Momentum of A before collision = M1*V1
The momentum of particle B is given by the mass (M2) of particle B multiplied by its initial velocity (V2):
Momentum of B before collision = M2*V2
The total momentum before the collision is the sum of the individual momenta:
Total momentum before collision = (M1*V1) + (M2*V2)
After the collision:
The momentum of particle A after the collision is given by the mass (M1) of particle A multiplied by its final velocity (V1):
Momentum of A after collision = M1*V1
The momentum of particle B after the collision is given by the mass (M2) of particle B multiplied by its final velocity (V2):
Momentum of B after collision = M2*V2
The total momentum after the collision is also the sum of the individual momenta:
Total momentum after collision = (M1*V1) + (M2*V2)
Since the principle of conservation of linear momentum states that the total momentum before and after the collision is the same, we can equate the two expressions:
(M1*V1) + (M2*V2) = (M1*V1) + (M2*V2)
Now let's solve for the final velocity of particle A (V1):
(M1*V1) + (M2*V2) = (M1*V1) + (M2*V2)
Rearranging the equation:
(M1*V1) - (M1*V1) = (M2*V2) - (M2*V2)
0 = 0
This result shows that there is no constraint on the final velocity of particle A (V1). Hence, the equation you mentioned is incorrect.
Please double-check the equation you provided or clarify the problem further if necessary.
To solve this problem, we can apply the principle of conservation of momentum. According to this principle, the total momentum before the collision is equal to the total momentum after the collision.
Before the collision, the momentum of particle A is given by P1 = M1 * V1, and the momentum of particle B is given by P2 = M2 * V2.
After the collision, the particle A's momentum changes to P1' = M1' * V1' (where V1' is the velocity of A after the collision), and the particle B's momentum changes to P2' = M2' * V2' (where V2' is the velocity of B after the collision).
Since the collision is perfectly elastic, the total kinetic energy before and after the collision should be equal. Therefore, we can write the following equation:
(1/2) * M1 * (V1)^2 + (1/2) * M2 * (V2)^2 = (1/2) * M1' * (V1')^2 + (1/2) * M2' * (V2')^2
Expanding this equation, we get:
(1/2) * M1 * (V1)^2 + (1/2) * M2 * (V2)^2 = (1/2) * M1' * (V1')^2 + (1/2) * M2' * (V2')^2
Multiplying each term by 2 to simplify the equation, we have:
M1 * (V1)^2 + M2 * (V2)^2 = M1' * (V1')^2 + M2' * (V2')^2
Now, using the principle of conservation of momentum, we know that the total momentum before the collision is equal to the total momentum after the collision. Therefore, we can write the following equation:
P1 + P2 = P1' + P2'
M1 * V1 + M2 * V2 = M1' * V1' + M2' * V2'
Since the collision is perfectly elastic, the velocities of the particles after the collision can be written as:
V1' = ((M1 - M2) / (M1 + M2)) * V1 + (2 * M2 / (M1 + M2)) * V2
V2' = ((2 * M1) / (M1 + M2)) * V1 + ((M2 - M1) / (M1 + M2)) * V2
Substituting these expressions into the conservation of momentum equation, we get:
M1 * V1 + M2 * V2 = ((M1 - M2) / (M1 + M2)) * V1 + (2 * M2 / (M1 + M2)) * V2 + ((2 * M1) / (M1 + M2)) * V1 + ((M2 - M1) / (M1 + M2)) * V2
Simplifying this equation, we obtain:
M1 * V1 + M2 * V2 = ((M1 - M2) * V1 + 2 * M2 * V2 + 2 * M1 * V1 + (M2 - M1) * V2) / (M1 + M2)
Multiplying each term by (M1 + M2) and simplifying further, we get:
M1 * V1 + M2 * V2 = M1 * V1 - M2 * V1 + 2 * M2 * V2 + 2 * M1 * V1 - M1 * V2 + M2 * V2
Canceling out like terms, we have:
M2 * V2 = -M2 * V1 + 2 * M1 * V1 - M1 * V2 + M2 * V2
Rearranging the terms, we obtain:
2 * M2 * V2 + M1 * V2 = M2 * V1 + 2 * M1 * V1
Factoring out V2 and V1, we have:
V2 * (2 * M2 + M1) = V1 * (M2 + 2 * M1)
Dividing both sides by (2 * M2 + M1), we obtain the final formula for V1:
V1 = (M2 - M1) / (M2 + M1) * V2 + 2 * M2 / (M2 + M1) * V1
To solve this problem, we can use the principle of conservation of momentum in collisions. According to this principle, the total momentum before the collision is equal to the total momentum after the collision.
Before the collision:
The momentum of particle A (M1) before the collision is given by:
Momentum of A before collision = M1 * V1
The momentum of particle B (M2) before the collision is given by:
Momentum of B before collision = M2 * V2
Total momentum before collision = Momentum of A before collision + Momentum of B before collision
Total momentum before collision = M1 * V1 + M2 * V2 --- (1)
After the collision:
Let V1' be the velocity of particle A after the collision.
The momentum of particle A (M1) after the collision is given by:
Momentum of A after collision = M1 * V1'
The momentum of particle B (M2) after the collision is given by:
Momentum of B after collision = M2 * V2 (as particle B remains at rest after the collision)
Total momentum after the collision = Momentum of A after collision + Momentum of B after collision
Total momentum after the collision = M1 * V1' + M2 * V2 --- (2)
According to the principle of conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision.
Therefore, equation (1) is equal to equation (2).
M1 * V1 + M2 * V2 = M1 * V1' + M2 * V2
To solve for V1', we can rearrange the equation as follows:
M1 * V1' = M1 * V1 + M2 * V2 - M2 * V2
= M1 * V1 + 0 (as M2 * V2 - M2 * V2 = 0)
M1 * V1' = M1 * V1
Dividing both sides of the equation by M1:
V1' = V1
Therefore, the velocity V1 of particle A after the impact is equal to the velocity V1 before the collision.