A 5kg object travelling at 0.1m/s collides head on with 10kg object initially at rest. Determine the velocity of each object after the impact if the collision is elastic
5 * 0.1 + 10 * 0 = 5 u + 10 v conservation of momentum
(1/2)(5)(0.1)^2 = (1/2)(5)(u^2) + (1/2)(10)(v^2) conservation of energy
Given:
M1 = 5kg, V1 = 0.1 m/s.
M2 = 10kg, Vz = 0.
V3 = Velocity of M1 after collision.
V4 = Velocity of M2 after collision.
Momentum before = Momentum after.
M1*V1 + M2*V2 = M1*V3 + M2*V4.
5 * 0.1 + 10 * 0 = 5*V3 + 10*V4,
Eq1: 5V3 + 10V4 = 0.5.
V3 = ((V1(M1-M2) + 2M2*V2)/(M1+M2).
V3 = ((0.1(5-10) + 20*0)/(5+10) = (-0.5 + 0)/15 = -0.0333 m/s. In opposite direction.
In Eq1, replace V3 with -0.0333 m/s and solve for V4.
To determine the velocity of each object after the impact in an elastic collision, we can use the principle of conservation of momentum.
The momentum before the collision is equal to the momentum after the collision. Mathematically, this can be written as:
(mass1 * velocity1)before + (mass2 * velocity2)before = (mass1 * velocity1)after + (mass2 * velocity2)after
Here,
mass1 = 5 kg (mass of the first object)
velocity1 before = 0.1 m/s (initial velocity of the first object)
mass2 = 10 kg (mass of the second object)
velocity2 before = 0 m/s (initial velocity of the second object)
velocity1 after = the final velocity of the first object after the impact
velocity2 after = the final velocity of the second object after the impact
Plugging in the given values, we get:
(5 kg * 0.1 m/s) + (10 kg * 0 m/s) = (5 kg * velocity1) + (10 kg * velocity2)
0.5 kg m/s = (5 kg * velocity1) + (10 kg * velocity2) ..(Equation 1)
Since the collision is elastic, both momentum and kinetic energy are conserved.
The equation for conservation of kinetic energy can be written as:
1/2 * mass1 * (velocity1 before)^2 + 1/2 * mass2 * (velocity2 before)^2 = 1/2 * mass1 * (velocity1 after)^2 + 1/2 * mass2 * (velocity2 after)^2
Plugging in the given values, we get:
1/2 * 5 kg * (0.1 m/s)^2 + 1/2 * 10 kg * (0 m/s)^2 = 1/2 * 5 kg * (velocity1 after)^2 + 1/2 * 10 kg * (velocity2 after)^2
0.025 J = 1.25 kg * (velocity1 after)^2 + 0 ..(Equation 2)
Now, we have two equations (Equation 1 and Equation 2) with two variables (velocity1 after and velocity2 after). We can solve these equations simultaneously to find the velocities.
First, let's solve Equation 1 for velocity1 after:
velocity1 after = (0.5 kg m/s - 10 kg * velocity2) / 5 kg
Substituting this value into Equation 2:
0.025 J = 1.25 kg * [(0.5 kg m/s - 10 kg * velocity2) / 5 kg]^2
Simplifying and solving for velocity2 after, we get:
velocity2 after = 0.03 m/s
Substituting this value back into the equation for velocity1 after:
velocity1 after = (0.5 kg m/s - 10 kg * 0.03 m/s) / 5 kg
velocity1 after = -0.1 m/s
Therefore, the final velocities of the two objects after the elastic collision are:
velocity1 after = -0.1 m/s (negative sign indicates that the first object is now moving in the opposite direction)
velocity2 after = 0.03 m/s
To determine the velocity of each object after the impact in an elastic collision, we need to apply the principle of conservation of momentum and conservation of kinetic energy.
The conservation of momentum states that the total momentum before the collision is equal to the total momentum after the collision.
The momentum (p) of an object is given by the product of its mass (m) and velocity (v), so we can write the initial momentum (p_initial) and final momentum (p_final) for the two objects as follows:
p_initial = m1 * v1 + m2 * v2 (1)
p_final = m1 * v1' + m2 * v2' (2)
where m1 and m2 are the masses of the two objects, v1 and v2 are their initial velocities, and v1' and v2' are their final velocities.
According to the conservation of momentum, p_initial = p_final, so we can equate Equation (1) and (2):
m1 * v1 + m2 * v2 = m1 * v1' + m2 * v2' (3)
Now, let's use the conservation of kinetic energy. In an elastic collision, the total kinetic energy before the collision is equal to the total kinetic energy after the collision.
The kinetic energy (KE) of an object is given by one-half times the product of its mass and the square of its velocity. So, we can write the initial kinetic energy (KE_initial) and final kinetic energy (KE_final) for the two objects as follows:
KE_initial = (1/2) * m1 * v1^2 + (1/2) * m2 * v2^2 (4)
KE_final = (1/2) * m1 * v1'^2 + (1/2) * m2 * v2'^2 (5)
According to the conservation of kinetic energy, KE_initial = KE_final, so we can equate Equation (4) and (5):
(1/2) * m1 * v1^2 + (1/2) * m2 * v2^2 = (1/2) * m1 * v1'^2 + (1/2) * m2 * v2'^2 (6)
We now have two equations (Equations 3 and 6) containing four unknowns (v1', v2', v1, v2). To solve for the unknowns in this system of equations, we need some additional information.
For example, if we know the initial velocity of one of the objects or the final velocity of one of the objects, we can solve for the remaining unknowns.