Hi, here's a problem that I have difficulty with to solve.
Block M2, w/c has a 20 kg mass, is on a frictionless table and it is firmly attached to one end of a spring with k = 2.5 N/m. The other end of the spring is anchored to the wall. Another block, M1 (and of the same mass as M2), is thrown horizontally toward M2 with a speed of 0.8 m/s.
For the elastic situation:
1) Calculate the final energy after the two masses collide
2) By how much is M2 compressed?
3) Describe the motion after M1 collides with M2
For inelastic situation:
1) Find the final velocity after the collision
2) Find the final momentum of the collision
To solve these problems, we'll need to use the principles of conservation of energy, conservation of momentum, and the equations of motion. Let's tackle each question step by step.
1) For the elastic situation, after the two masses collide:
To calculate the final energy, we need to consider the initial kinetic energy (KE) of M1 and the final potential energy (PE) stored in the spring.
- The initial KE of M1 is given by the formula KE = (1/2) * m * v^2, where m is the mass of M1 and v is its initial velocity. So, KE = (1/2) * 20 kg * (0.8 m/s)^2.
After the collision, all the kinetic energy is converted into potential energy stored in the spring. The formula for the potential energy of a spring is PE = (1/2) * k * x^2, where k is the spring constant and x is the compression of the spring.
- So, the final energy after the collision is the potential energy of the spring. Let's solve for x in the formula: PE = (1/2) * k * x^2.
PE = (1/2) * 2.5 N/m * x^2.
x^2 = 2 PE / k.
x = sqrt(2 PE / k).
2) To find how much M2 is compressed:
Using the equation derived in the previous step, plug in the values we know: m = 20 kg, k = 2.5 N/m, and the calculated PE from step 1.
- Rearranging the formula, we have: x = sqrt(2 * PE / k).
x = sqrt(2 * PE / 2.5).
x = sqrt(0.8 * PE).
3) To describe the motion after M1 collides with M2:
Assuming M2 is initially at rest and M1 collides horizontally with M2:
- M1 will transfer its momentum to M2. Since both masses are equal (20 kg) and initially have the same speed, the final velocities will be equal in magnitude but opposite in direction.
- Therefore, M2 will move in the opposite direction but with the same speed as M1 had initially.
For the inelastic situation, we need to use the principles of conservation of momentum:
1) To find the final velocity after the collision:
Applying the conservation of momentum, the total momentum before the collision equals the total momentum after the collision.
- The initial momentum is given by the formula p = m * v, where m is the mass and v is the velocity. Thus, the initial momentum of both masses is p = 20 kg * 0.8 m/s.
- After the collision, they stick together and move as a single mass. Since they stick together, their masses get added, resulting in a final mass of 40 kg.
- We can use the equation p = m * v to find the final velocity. The initial momentum equals the final momentum: 20 kg * 0.8 m/s = 40 kg * v.
2) To find the final momentum of the collision:
The final momentum can be calculated using the same equation p = m * v, but this time with the final velocity found in step 1.
- The final momentum is given by p = 40 kg * v.
And that's how we can solve the given problems using the principles of conservation of energy and momentum, as well as the equations of motion.