Two blocks move along a linear path on a nearly frictionless air track. One block, of mass 0.120 kg, initially moves to the right at a speed of 4.90 m/s, while the second block, of mass 0.240 kg, is initially to the left of the first block and moving to the right at 7.40 m/s. Find the final velocities of the blocks, assuming the collision is elastic.
To find the final velocities of the blocks after an elastic collision, we can use the principles of conservation of momentum and kinetic energy.
1. Conservation of Momentum: In an elastic collision, the total momentum before and after the collision is conserved. We can write this as:
m1 * v1initial + m2 * v2initial = m1 * v1final + m2 * v2final
where m1 and m2 are the masses of the first and second blocks respectively, v1 and v2 are their velocities before and after the collision.
2. Conservation of Kinetic Energy: In an elastic collision, the total kinetic energy before and after the collision is conserved. We can write this as:
0.5 * m1 * v1initial^2 + 0.5 * m2 * v2initial^2 = 0.5 * m1 * v1final^2 + 0.5 * m2 * v2final^2
Now, let's solve these equations step by step:
Given data:
m1 = 0.120 kg (mass of first block)
v1initial = 4.90 m/s (initial velocity of first block)
m2 = 0.240 kg (mass of second block)
v2initial = -7.40 m/s (initial velocity of second block)
Step 1: Calculate the momentum before the collision:
momentum_initial = m1 * v1initial + m2 * v2initial
Step 2: Apply conservation of momentum to find the final velocities:
m1 * v1initial + m2 * v2initial = m1 * v1final + m2 * v2final
=> v1final + v2final = (m1 * v1initial + m2 * v2initial) / (m1 + m2)
Step 3: Calculate the kinetic energy before the collision:
kinetic_energy_initial = 0.5 * m1 * v1initial^2 + 0.5 * m2 * v2initial^2
Step 4: Apply conservation of kinetic energy to find the final velocities:
0.5 * m1 * v1initial^2 + 0.5 * m2 * v2initial^2 = 0.5 * m1 * v1final^2 + 0.5 * m2 * v2final^2
Step 5: Solve the two equations to find the final velocities:
Simplify the equations and substitute the values obtained in Step 1 and Step 2.
By solving these equations, you can find the final velocities of the blocks.
To solve this problem, we can apply the principles of conservation of momentum and kinetic energy.
Step 1: Calculate the initial momentum of each block.
The momentum of an object is defined as the product of its mass and velocity (p = mv).
For the first block:
Initial momentum = mass x velocity = 0.120 kg x 4.90 m/s = 0.588 kg·m/s
For the second block:
Initial momentum = mass x velocity = 0.240 kg x (-7.40 m/s) = -1.776 kg·m/s (negative because it's moving in the opposite direction)
Step 2: Calculate the total initial momentum of the system.
Since momentum is a vector quantity, the total initial momentum of the system (both blocks together) can be found by adding the momenta of each block.
Total initial momentum = momentum of 1st block + momentum of 2nd block
Total initial momentum = 0.588 kg·m/s + (-1.776 kg·m/s) = -1.188 kg·m/s
Step 3: Apply the principle of conservation of momentum to find the final velocities of the blocks.
In an elastic collision, the total momentum before the collision is equal to the total momentum after the collision.
Total initial momentum = Total final momentum
-1.188 kg·m/s = (mass of 1st block x velocity of 1st block) + (mass of 2nd block x velocity of 2nd block)
Let's assume the final velocities of the blocks are Vf1 and Vf2.
-1.188 kg·m/s = (0.120 kg x Vf1) + (0.240 kg x Vf2)
Step 4: Apply the principle of conservation of kinetic energy to find the final velocities of the blocks.
In an elastic collision, the total kinetic energy before the collision is equal to the total kinetic energy after the collision.
Total initial kinetic energy = Total final kinetic energy
The kinetic energy of an object is given by the formula (KE = 0.5 * mass * velocity^2).
Total initial kinetic energy = (0.5 x mass of 1st block x (velocity of 1st block)^2) + (0.5 x mass of 2nd block x (velocity of 2nd block)^2)
Total initial kinetic energy = (0.5 x 0.120 kg x (4.90 m/s)^2) + (0.5 x 0.240 kg x (7.40 m/s)^2)
Total initial kinetic energy = 0.2946 J + 0.80832 J
Total initial kinetic energy = 1.10292 J
Total final kinetic energy = (0.5 x mass of 1st block x (Vf1)^2) + (0.5 x mass of 2nd block x (Vf2)^2)
Step 5: Solve the simultaneous equations.
We have two equations: one for conservation of momentum and one for conservation of kinetic energy. We can solve these equations simultaneously to find the final velocities of the blocks.
-1.188 kg·m/s = (0.120 kg x Vf1) + (0.240 kg x Vf2) (Equation 1)
1.10292 J = (0.5 x 0.120 kg x (Vf1)^2) + (0.5 x 0.240 kg x (Vf2)^2) (Equation 2)
Solving these equations will give the values of Vf1 and Vf2, which are the final velocities of the blocks.