A 0.43 kg object connected to a light spring with a spring constant of 21.4 N/m oscillates on a frictionless horizontal surface. The spring is compressed 4.0 cm and released from rest.

(a) Determine the maximum speed of the mass.
(b) Determine the speed of the mass when the spring is compressed 1.5 cm.
(c) Determine the speed of the mass when the spring is stretched 1.5 cm.
(d) For what value of x does the speed equal one-half the maximum speed?

To solve this problem, we can apply the principles of simple harmonic motion (SHM) and the formulas related to it. Let's go through each part of the question:

(a) To determine the maximum speed of the mass, we can use the formula for the maximum velocity in SHM: vmax = Aω, where A is the amplitude (maximum displacement) and ω is the angular frequency.

Since the spring is compressed by 4.0 cm, the amplitude in this case, A, would be 0.04 m. The angular frequency ω can be calculated from the spring constant, k, using the formula ω = √(k/m), where m is the mass of the object connected to the spring.

Given that k = 21.4 N/m and m = 0.43 kg, we can substitute these values into the formula to find ω. Then, we can calculate vmax using the formula mentioned above.

(b) To determine the speed of the mass when the spring is compressed 1.5 cm, we need to find the displacement of the mass from its equilibrium position. The displacement can be calculated as x = A - d, where d is the distance the spring is compressed or stretched from its equilibrium position.

Using the same formula as before, we can find the angular frequency ω. Then, we can calculate the velocity v using the formula v = Aω.

(c) To determine the speed of the mass when the spring is stretched 1.5 cm, we follow the same procedure as in part (b). In this case, the displacement x would be x = A + d.

(d) To find the value of x where the speed equals one-half the maximum speed, we need to use the equation for the speed in SHM: v = ω√(A^2 - x^2). Substitute v = vmax/2 into this equation and solve for x.

Please provide me with the values of k, m, and d, and I'll help you calculate the answers to parts (a) to (d).