A block of mass m1 = 2.4 kg initially moving to the right with a speed of 3.3 m/s on a frictionless, horizontal track collides with a spring attached to a second block of mass m2 = 4.1 kg initially moving to the left with a speed of
1.5
m/s as shown in figure (a). The spring constant is 514 N/m.
What will be the individual velocities of the two masses (v1 and v2) after the spring extended fully again?
To determine the individual velocities of the two masses, we can apply the principles of conservation of momentum and conservation of energy.
First, let's calculate the initial total momentum of the system before the collision:
Initial total momentum = momentum of block 1 + momentum of block 2
= (mass of block 1 * velocity of block 1) + (mass of block 2 * velocity of block 2)
Initial total momentum = (2.4 kg * 3.3 m/s) + (4.1 kg * (-1.5 m/s))
= 7.92 kg·m/s - 6.15 kg·m/s
= 1.77 kg·m/s
Next, let's determine the total potential energy stored in the spring when it is fully extended. The potential energy stored in a spring can be calculated using the formula:
Potential energy = (1/2) * spring constant * extension^2
In this case, the extension is the displacement of the spring from its equilibrium position. Since the spring is fully extended, the extension is the maximum displacement before compression. Therefore, the potential energy stored in the spring is at its maximum.
Potential energy = (1/2) * 514 N/m * extension^2
To find the extension, we can use the principle of conservation of energy. Initially, the system has only kinetic energy, given by:
Initial kinetic energy = (1/2) * mass of block 1 * (velocity of block 1)^2 + (1/2) * mass of block 2 * (velocity of block 2)^2
Initial kinetic energy = (1/2) * 2.4 kg * (3.3 m/s)^2 + (1/2) * 4.1 kg * (1.5 m/s)^2
By conservation of energy, the initial kinetic energy should be equal to the maximum potential energy stored in the spring when fully extended:
Initial kinetic energy = Potential energy
Substituting the values, we have:
(1/2) * 2.4 kg * (3.3 m/s)^2 + (1/2) * 4.1 kg * (1.5 m/s)^2 = (1/2) * 514 N/m * extension^2
Solving this equation will provide the value of the extension.
Once we have the extension, we can calculate the compression in the spring during the collision by subtracting the initial extension from the maximum extension.
Finally, we can use conservation of momentum to find the final velocities of the two masses. Knowing the compression in the spring and the forces acting on the masses, we can relate the change in momentum to the change in potential energy stored in the spring.
To summarize the process to find the individual velocities of the two masses:
1. Calculate the initial total momentum of the system.
2. Calculate the initial kinetic energy of the system.
3. Set the initial kinetic energy equal to the potential energy stored in the spring when fully extended.
4. Solve for the extension of the spring.
5. Calculate the compression in the spring during the collision.
6. Use conservation of momentum to find the final velocities of the two masses.