A solid block of mass m2 = 8.3 kg, at rest on a horizontal frictionless surface, is connected to a relaxed spring. The other end of the spring is fixed, and the spring constant is k = 210 N/m. Another solid block of mass m1 = 12.4 kg and speed v1 = 4.9 m/s collides with the 8.3 kg block. The blocks stick together, and compress the spring. What is the maximum compression of the spring? What is the magnitude of the acceleration of the two blocks when the spring is at maximum compression?
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To solve this problem, we need to apply the principles of conservation of momentum and conservation of mechanical energy.
1. Determine the initial momentum of the system:
The total initial momentum of the system is the sum of the momenta of the two blocks. The momentum (p) is given by the product of mass (m) and velocity (v). So, the initial momentum is:
p_initial = (m1 * v1) + (m2 * 0) (since the second block is at rest initially)
= m1 * v1
2. Determine the final velocity of the combined blocks:
After the collision, the two blocks stick together. Since momentum is conserved, the total momentum of the system after the collision should be equal to the initial momentum. Thus:
p_final = (m1 + m2) * vf
Setting p_initial equal to p_final, we have:
m1 * v1 = (m1 + m2) * vf
Now, solve for the final velocity (vf).
3. Calculate the maximum compression of the spring:
The maximum compression of the spring occurs when all the initial kinetic energy of the block m1 is transferred into the potential energy stored in the spring. At maximum compression, the blocks momentarily come to rest, and all the initial kinetic energy is converted into the potential energy of the spring.
Use the principle of conservation of mechanical energy:
Kinetic energy before collision = Potential energy of the spring at maximum compression
The initial kinetic energy (KE_initial) is given by:
KE_initial = 0.5 * m1 * v1^2
The potential energy of the spring at maximum compression (PE_spring) is given by:
PE_spring = 0.5 * k * x^2 (where x is the maximum compression)
Setting KE_initial equal to PE_spring and solving for x, we can find the maximum compression of the spring.
4. Find the magnitude of acceleration at maximum compression:
The acceleration (a) of the two blocks at maximum compression can be calculated using Hooke's law, which states that the force exerted by a spring is proportional to the displacement from its equilibrium position:
F = -k * x (negative sign indicates the force is in the opposite direction of the displacement)
Since F = m * a (Newton's second law), we can set the expressions for force equal to mass times acceleration:
-m1 * a = -k * x
After solving for a, we can determine the magnitude of acceleration at maximum compression.
By following these steps, you can find the maximum compression of the spring and the magnitude of acceleration for the given scenario.