A 0.70 kg block along a horizontal track has a speed of 1.10 m/s immediately before colliding with a light spring of force constant 3.50 N/m located at the end of the track. (a) What is the spring’s maximum compression if the track is frictionless?

mv²/2 = kx²/2

x=sqrt(mv²/k)

.492

To find the spring's maximum compression, we can use the principle of conservation of mechanical energy.

Step 1: Determine the initial kinetic energy of the block.
The initial kinetic energy (KE) of the block can be calculated using the formula:
KE_initial = (1/2) * m * v_initial^2
where m is the mass of the block (0.70 kg) and v_initial is the initial speed of the block (1.10 m/s).
Substituting the values:
KE_initial = (1/2) * 0.70 kg * (1.10 m/s)^2 = 0.4235 J

Step 2: Determine the maximum potential energy stored in the spring.
The maximum potential energy (PE) stored in the spring is equal to the initial kinetic energy of the block.
PE_spring = KE_initial = 0.4235 J

Step 3: Use the formula for potential energy stored in a spring.
The potential energy stored in a spring is given by the formula:
PE_spring = (1/2) * k * x^2
where k is the force constant of the spring (3.50 N/m) and x is the maximum compression of the spring.
Rearranging the equation:
x^2 = (2 * PE_spring) / k = (2 * 0.4235 J) / 3.50 N/m
x^2 = 0.242 J/m

Step 4: Calculate the maximum compression of the spring.
Taking the square root of both sides of the equation:
x = √(0.242 J/m) ≈ 0.492 m

Therefore, the spring's maximum compression is approximately 0.492 meters.

To determine the spring's maximum compression, we can apply the principle of conservation of mechanical energy. This principle states that the total mechanical energy of a system remains constant if no external forces do work on the system.

Initially, the block has only kinetic energy, given by the equation:

KE_initial = (1/2) * m * v^2

where m is the mass of the block and v is its initial velocity. In this case, m = 0.70 kg and v = 1.10 m/s. Thus:

KE_initial = (1/2) * 0.70 kg * (1.10 m/s)^2

Next, as the block collides with the spring, its kinetic energy is gradually converted into potential energy stored in the compressed spring. At the spring's maximum compression, all the block's kinetic energy is transformed into potential energy:

PE_spring = (1/2) * k * x^2

where k is the spring constant and x is the spring's compression. In this case, k = 3.50 N/m. By equating the initial kinetic energy and the potential energy at maximum compression, we can solve for x:

KE_initial = PE_spring

(1/2) * m * v^2 = (1/2) * k * x^2

0.35 kg * (1.10 m/s)^2 = (1/2) * 3.50 N/m * x^2

0.35 kg * 1.21 m^2 = 1.75 N/m * x^2

0.4235 Ns^2/m = x^2

Taking the square root of both sides, we find:

x = √(0.4235 Ns^2/m)

x ≈ 0.651 m

Therefore, the spring's maximum compression, in the absence of friction, is approximately 0.651 meters.