Two bodies a and b masses 4kg and 2kg move towards each other with velocities 3m/s and 2m/s and collide. If the collision is perfectly inelastic, find the velocity
of the 2 bodies after collision. Find the tolal kinetic energy of the system before and after collision, hence calculate the loss in kinetic energy.
To find the velocity of the two bodies after the perfectly inelastic collision, we can use the conservation of momentum principle.
The formula for the conservation of momentum is:
m₁v₁ + m₂v₂ = (m₁ + m₂)v
Where m₁ and m₂ are the masses of bodies a and b respectively, v₁ and v₂ are their initial velocities, and v is their final velocity.
Given:
Mass of body a (m₁) = 4 kg
Mass of body b (m₂) = 2 kg
Initial velocity of body a (v₁) = 3 m/s
Initial velocity of body b (v₂) = 2 m/s
Substituting the given values into the momentum conservation equation:
(4 kg * 3 m/s) + (2 kg * 2 m/s) = (4 kg + 2 kg) * v
12 kgm/s + 4 kgm/s = 6 kg * v
16 kgm/s = 6 kgv
Dividing both sides by 6 kg:
v = 16 kgm/s / 6 kg
v ≈ 2.67 m/s
Therefore, the velocity of the two bodies after the collision is approximately 2.67 m/s.
To find the total kinetic energy of the system before and after the collision, we can use the formula:
Kinetic Energy = 1/2 * mass * velocity^2
For the system before the collision:
Total initial kinetic energy = (1/2 * 4 kg * (3 m/s)^2) + (1/2 * 2 kg * (2 m/s)^2)
= (1/2 * 4 kg * 9 m^2/s^2) + (1/2 * 2 kg * 4 m^2/s^2)
= 18 J + 4 J
= 22 J
For the system after the collision:
Using the final velocity (v ≈ 2.67 m/s) from the previous calculation:
Total final kinetic energy = (1/2 * 6 kg * (2.67 m/s)^2)
= 8 J
Therefore, the total kinetic energy before the collision is 22 J, and the total kinetic energy after the collision is 8 J.
The loss in kinetic energy can be calculated as:
Loss in kinetic energy = Total initial kinetic energy - Total final kinetic energy
= 22 J - 8 J
= 14 J
Therefore, the loss in kinetic energy is 14 J.
To find the velocity of the two bodies after collision, we can apply the principle of conservation of momentum.
Conservation of momentum states that the total momentum before a collision is equal to the total momentum after the collision.
The momentum of an object is defined as the product of its mass and velocity. Mathematically, momentum (p) is given by p = m * v, where p is momentum, m is mass, and v is velocity.
Let's calculate the total momentum before the collision:
Initial momentum of body A = mass of A * velocity of A = 4 kg * 3 m/s = 12 kg⋅m/s
Initial momentum of body B = mass of B * velocity of B = 2 kg * (-2 m/s) = -4 kg⋅m/s (negative sign indicates opposite direction)
Now, since the collision is perfectly inelastic, the two bodies stick together after the collision and move as a single mass. Let the combined mass after the collision be M, and the velocity of the combined mass be V.
The total momentum after the collision is the momentum of the combined mass:
Final momentum of the combined mass = M * V
According to the conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision:
Initial momentum of body A + Initial momentum of body B = Final momentum of the combined mass
12 kg⋅m/s - 4 kg⋅m/s = M * V
Using the given information, we can solve for V:
8 kg⋅m/s = M * V
Now, to find the combined mass, we add the individual masses of the two bodies:
Combined mass after collision, M = mass of A + mass of B = 4 kg + 2 kg = 6 kg
Plugging in the values, we can solve for V:
8 kg⋅m/s = 6 kg * V
V = 8 kg⋅m/s / 6 kg
V = 4/3 m/s
So, the velocity of the two bodies after the perfectly inelastic collision is 4/3 m/s.
Now, let's calculate the total kinetic energy of the system before and after the collision:
The kinetic energy of an object is given by the formula KE = (1/2) * m * v^2, where KE is kinetic energy, m is mass, and v is velocity.
Total kinetic energy before the collision:
KE_before = (1/2) * mass of A * (velocity of A)^2 + (1/2) * mass of B * (velocity of B)^2
= (1/2) * 4 kg * (3 m/s)^2 + (1/2) * 2 kg * (2 m/s)^2
= 18 J + 4 J
= 22 J
Total kinetic energy after the collision:
KE_after = (1/2) * mass of combined mass * (velocity of the combined mass)^2
= (1/2) * 6 kg * (4/3 m/s)^2
= 8 J
The loss in kinetic energy can be calculated by subtracting the total kinetic energy after the collision from the total kinetic energy before the collision:
Loss in kinetic energy = KE_before - KE_after
= 22 J - 8 J
= 14 J
Therefore, the loss in kinetic energy during the perfectly inelastic collision is 14 J.
If it is perfectly inelastic, no bounce at all, then they are stuck together after the collision and all the momentum is in one 6 kg mass
4*3 - 2*2 = 6 * v
before ke = (1/2)(4)(9) + (1/2)(2)(4)
after ke = (1/2)(6)v^2