A 0.290-kg block along a horizontal track has a speed of 1.30 m/s immediately before colliding with a light spring of force constant 53.4 N/m located at the end of the track.

(a) What is the spring's maximum compression if the track is frictionless?
m

KE=1/2 K x^2

x=.290(1.3)^2/ 53.4 meters

To find the spring's maximum compression, we can use the concepts of energy conservation and Hooke's law.

Let's start by finding the initial kinetic energy of the block. The formula for kinetic energy is:
KE = (1/2) * m * v^2

where:
KE is the kinetic energy,
m is the mass of the block,
v is the speed of the block.

Substituting the given values:
m = 0.290 kg
v = 1.30 m/s

KE = (1/2) * 0.290 kg * (1.30 m/s)^2
KE = 0.290 kg * 0.845 m^2/s^2
KE = 0.245 m^2kg/s^2

Now, when the block collides with the spring, the energy is transferred into potential energy of the compressed spring. According to Hooke's law, the potential energy in a spring is given by:
PE = (1/2) * k * x^2

where:
PE is the potential energy,
k is the force constant of the spring,
x is the displacement (compression) of the spring from its equilibrium position.

Since energy is conserved, the initial kinetic energy of the block is equal to the potential energy stored in the compressed spring. Therefore, we can set up the equation:

KE = PE
0.245 m^2kg/s^2 = (1/2) * 53.4 N/m * x^2

Simplifying the equation:
0.245 = 26.7 * x^2

To find the maximum compression (x), we can rearrange the equation and solve for x:

x^2 = 0.245 / 26.7
x^2 = 0.00916

Taking the square root of both sides:
x = 0.0957 m

Therefore, the spring's maximum compression is approximately 0.0957 m.