Two blocks move along a linear path on a nearly frictionless air track. One block, of mass 0.104 kg, initially moves to the right at a speed of 4.70 m/s, while the second block, of mass 0.208 kg, is initially to the left of the first block and moving to the right at 6.50 m/s. Find the final velocities of the blocks, assuming the collision is elastic
To solve this problem, we can apply the principles of conservation of momentum and conservation of kinetic energy for an elastic collision.
Step 1: Find the initial total momentum (Pi) of the system.
The momentum of an object is given by the equation:
Momentum = mass * velocity
For the first block:
Mass1 = 0.104 kg
Velocity1 = 4.70 m/s
Momentum1 = Mass1 * Velocity1
For the second block:
Mass2 = 0.208 kg
Velocity2 = -6.50 m/s (since the second block is moving in the opposite direction)
Momentum2 = Mass2 * Velocity2
The initial total momentum (Pi) of the system is the sum of the individual momenta of the blocks:
Pi = Momentum1 + Momentum2
Step 2: Find the initial total kinetic energy (Ki) of the system.
The kinetic energy of an object is given by the equation:
Kinetic energy = (1/2) * mass * (velocity^2)
For the first block:
Kinetic energy1 = (1/2) * Mass1 * (Velocity1^2)
For the second block:
Kinetic energy2 = (1/2) * Mass2 * (Velocity2^2)
The initial total kinetic energy (Ki) of the system is the sum of the individual kinetic energies of the blocks:
Ki = Kinetic energy1 + Kinetic energy2
Step 3: Apply the conservation of momentum and conservation of kinetic energy equations for an elastic collision.
When two objects collide elastically, the total momentum and total kinetic energy before the collision are equal to the total momentum and total kinetic energy after the collision.
Conservation of momentum:
Pi = Pf, where Pf is the final total momentum of the system.
Conservation of kinetic energy:
Ki = Kf, where Kf is the final total kinetic energy of the system.
Step 4: Solve the equations.
Based on the conservation of momentum:
Pi = Pf
Momentum1 + Momentum2 = Momentum1' + Momentum2', where the primes (') denote the final velocities.
Based on the conservation of kinetic energy:
Ki = Kf
Kinetic energy1 + Kinetic energy2 = Kinetic energy1' + Kinetic energy2', where the primes (') denote the final kinetic energies.
Using the formulas for momentum and kinetic energy, substitute the given values into these equations:
(Mass1 * Velocity1) + (Mass2 * Velocity2) = (Mass1 * Velocity1') + (Mass2 * Velocity2')
(1/2) * Mass1 * (Velocity1^2) + (1/2) * Mass2 * (Velocity2^2) = (1/2) * Mass1 * (Velocity1'^2) + (1/2) * Mass2 * (Velocity2'^2)
Now, we need to solve these equations to find the final velocities of the blocks. The calculations involve quadratic equations.
To find the final velocities of the blocks after the collision, we can use the principles of conservation of momentum and kinetic energy.
First, let's find the initial momentum of each block (before the collision). Momentum is defined as the product of an object's mass and velocity:
Momentum of the first block (m1):
m1 = 0.104 kg
v1 (initial velocity of the first block) = 4.70 m/s
Momentum of the first block (p1) = m1 * v1
Momentum of the second block (m2):
m2 = 0.208 kg
v2 (initial velocity of the second block) = 6.50 m/s
Momentum of the second block (p2) = m2 * v2
The total initial momentum (p_initial) is the sum of the individual momenta of the blocks:
p_initial = p1 + p2
Now, let's apply the principle of conservation of momentum. According to this principle, the total momentum before the collision should be equal to the total momentum after the collision:
p_initial = p_final
Since the collision is elastic, the kinetic energy is conserved as well. We can use this information to solve for the final velocities of the blocks.
The formula for kinetic energy (KE) is given by:
KE = 0.5 * mass * (velocity)^2
Let's denote the final velocities of the first and second blocks as v1' and v2', respectively.
The total initial kinetic energy (KE_initial) is the sum of the individual kinetic energies of the blocks:
KE_initial = 0.5 * m1 * v1^2 + 0.5 * m2 * v2^2
The total final kinetic energy (KE_final) is the sum of the individual kinetic energies of the blocks after the collision:
KE_final = 0.5 * m1 * (v1')^2 + 0.5 * m2 * (v2')^2
Since the kinetic energy is conserved, we can set KE_initial equal to KE_final:
0.5 * m1 * v1^2 + 0.5 * m2 * v2^2 = 0.5 * m1 * (v1')^2 + 0.5 * m2 * (v2')^2
Now, we have a system of two equations - one equation for the conservation of momentum and another equation for the conservation of kinetic energy.
By solving these two equations simultaneously, we can find the final velocities v1' and v2' of the blocks after the collision.