A block of mass m1 = 4.00 kg slides along a frictionless table with a speed of 16.0 m/s. Directly in front of it, and moving in the same direction, is a block of mass m2 = 5.00 kg moving at 7.00 m/s. A massless spring with a spring constant k = 1220 N/m is attached to the backside of m2. When the blocks collide, what is the maximum compression (in meters) of the spring?
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To find the maximum compression of the spring during the collision, we can use the principles of conservation of momentum and conservation of mechanical energy.
Let's break down the problem into steps:
Step 1: Calculate the initial momentum of each block.
The momentum of an object is given by the product of its mass and velocity.
For block m1:
Momentum = mass * velocity = m1 * v1
where m1 = 4.00 kg and v1 = 16.0 m/s.
For block m2:
Momentum = mass * velocity = m2 * v2
where m2 = 5.00 kg and v2 = 7.00 m/s.
Step 2: Determine the total initial momentum.
Since the two blocks are moving in the same direction, their momenta add up to give the total initial momentum.
Initial momentum = momentum of m1 + momentum of m2
Step 3: Calculate the final momentum after the collision.
According to the law of conservation of momentum, the total momentum before and after the collision remains the same. Therefore, the final momentum will also be equal to the total initial momentum.
Step 4: Determine the final velocity of the blocks after the collision.
Since the two blocks are connected by a spring, they will move together after the collision. This means they will have the same final velocity, denoted as vf.
Step 5: Apply the conservation of mechanical energy.
The energy of the system is conserved since there is no external work done on the system. In this problem, the mechanical energy is in the form of kinetic energy before the collision and elastic potential energy after the collision.
The total mechanical energy before the collision is given by the sum of the kinetic energies of both blocks:
Initial mechanical energy = (1/2) * m1 * v1^2 + (1/2) * m2 * v2^2
The total mechanical energy after the collision is given by the elastic potential energy stored in the spring when it is compressed:
Final mechanical energy = (1/2) * k * x^2
where k = 1220 N/m is the spring constant and x is the maximum compression of the spring.
Since the mechanical energy remains constant, we can set the initial mechanical energy equal to the final mechanical energy:
Initial mechanical energy = Final mechanical energy
(1/2) * m1 * v1^2 + (1/2) * m2 * v2^2 = (1/2) * k * x^2
Step 6: Solve for x (maximum compression of the spring).
Rearrange and solve for x:
x^2 = (m1 * v1^2 + m2 * v2^2) / k
x = √[(m1 * v1^2 + m2 * v2^2) / k]
Now substitute the given values of m1, v1, m2, v2, and k into the above equation and calculate the maximum compression of the spring, x.