Two bodies having masses 20kg and 10kg are moving with uniform velocities 20m/sec and 10m/sec from opposite direction and make elastic collision. find the final velocities of two bodies.
8m/sec and 2m/sec
Please give me the solution
initial x momentum = 20*20-10*10 =+300
initial Ke=.5*20*400 +.5*10*100=4500
final speeds in x direction are u and v
so
20 u + 10 v = +300 so v=(30-2u)
.5(20)(u^2) + .5(10)(v^2) = +4500
so
10u^2 +5(900-120u+4u^2) = 4500
u^2 + 450 - 60 u + 2 u^2 = 450
3 u^2 -60 u = 0
u = 0 then v = +30
or the initial case (trivial)
or u = 20
then v = - 10
so I think 0 and +30
To find the final velocities of the two bodies after an elastic collision, we can use 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.
The momentum of an object is given by the product of its mass and velocity. So, let's calculate the initial momentum of the two bodies before the collision.
Initial momentum of the 20kg body = mass * velocity = 20kg * 20m/sec = 400kgm/sec.
Initial momentum of the 10kg body = mass * velocity = 10kg * (-10m/sec) = -100kgm/sec (since the velocity is in the opposite direction).
The total initial momentum before the collision is the sum of the individual momenta.
Total initial momentum = 400kgm/sec + (-100kgm/sec) = 300kgm/sec.
Now, let's consider the final velocities of the two bodies after the collision. Since this is an elastic collision, the total kinetic energy before the collision is equal to the total kinetic energy after the collision.
The kinetic energy of an object is given by (1/2) * mass * velocity^2.
Let's calculate the initial kinetic energy before the collision.
Initial kinetic energy of the 20kg body = (1/2) * 20kg * (20m/sec)^2 = 2000J.
Initial kinetic energy of the 10kg body = (1/2) * 10kg * (10m/sec)^2 = 500J.
The total initial kinetic energy before the collision is the sum of the individual kinetic energies.
Total initial kinetic energy = 2000J + 500J = 2500J.
Now, let's find the final velocities.
After the collision, the two bodies exchange momentum and energy, but the total momentum and total kinetic energy remain the same.
According to the principle of conservation of momentum:
Total final momentum = Total initial momentum,
which means:
Momentum of the 20kg body + Momentum of the 10kg body = Total initial momentum.
Let's assume the final velocities of the 20kg and 10kg bodies are v1 and v2, respectively.
Final momentum of the 20kg body = mass * velocity = 20kg * v1.
Final momentum of the 10kg body = mass * velocity = 10kg * v2.
Since the two bodies are moving in opposite directions, we'll need to consider the directions explicitly:
The final momentum of the 20kg body is in the positive direction, and the final momentum of the 10kg body is in the negative direction.
So:
20kg * v1 + (-10kg * v2) = 300kgm/sec.
Now, let's consider the principle of conservation of kinetic energy:
Total final kinetic energy = Total initial kinetic energy,
which means:
Kinetic energy of the 20kg body + Kinetic energy of the 10kg body = Total initial kinetic energy.
Final kinetic energy of the 20kg body = (1/2) * 20kg * (v1)^2.
Final kinetic energy of the 10kg body = (1/2) * 10kg * (v2)^2.
So:
(1/2) * 20kg * (v1)^2 + (1/2) * 10kg * (v2)^2 = 2500J.
Now, we have two equations:
20kg * v1 + (-10kg * v2) = 300kgm/sec, ---(Equation 1)
(1/2) * 20kg * (v1)^2 + (1/2) * 10kg * (v2)^2 = 2500J. ---(Equation 2)
We can solve these two equations simultaneously to find the values of v1 and v2, which are the final velocities of the 20kg and 10kg bodies, respectively.