Momentum Problem: Masses of 4kg & 6kg, K = 100n/m, 10m/sec

[Anzas]

a mass of 4kg is tied to a spring, the spring's constant is K = 100n/m
a second mass of 6kg is moving towards the spring at a speed of 10m/sec

1.what is the spring's shrinking.
2.what is the final speed of both masses.

 

[futb0l]

1. Conservation of Energy

$$\frac{1}{2}m_1v_1^2 = \frac{1}{2}kx^2$$

The kinetic energy of the moving mass is converted into the strain energy of the spring. This energy is then converted into kinetic energy of the two masses.

2. Assign the right direction as positive. Now, you have two equations, conservation of momentum and conservation of energy, so assuming no external forces and energy loss, it's going to be just like an elastic collision.

I found that:

$$v_1' = 2 m/s$$
$$v_2' = 12 m/s$$

Check it for yourself, I might've got it completely wrong.

 

[thepatientmental]

1. The spring's shrinking can be calculated using the formula for potential energy stored in a spring, which is given by U = 1/2kx^2, where U is the potential energy, k is the spring constant, and x is the displacement of the spring. In this case, the displacement of the spring is equal to the initial velocity of the 6kg mass, which is 10m/sec. Thus, the potential energy stored in the spring is U = 1/2(100)(10)^2 = 5000 J. This potential energy will be converted into kinetic energy as the spring shrinks, and this kinetic energy will be transferred to the 4kg mass.

2. To calculate the final speed of both masses, we can use the principle of conservation of momentum, which states that the total momentum of a closed system remains constant. In this case, the initial momentum of the system is equal to the momentum of the 6kg mass, which is given by P = mv = (6)(10) = 60 kg-m/s. As the spring shrinks and transfers its energy to the 4kg mass, the final momentum of the system will be equal to the sum of the momentums of both masses. Thus, we can set up the equation: P = (4)(vf) + (6)(10), where vf is the final velocity of the 4kg mass. Solving for vf, we get vf = (60-60)/4 = 0 m/s. This means that the 4kg mass will come to a stop, while the 6kg mass will continue moving at 10m/sec. Therefore, the final speed of the 6kg mass will be 10m/sec, and the final speed of the 4kg mass will be 0m/sec.