Final answer:
The spring's Elastic Potential Energy (EPE) is 13.5 J; this is fully converted into the box's Kinetic Energy (KE) when the spring expands. The box moves at approximately 1.16 m/s after the spring releases it.
Step-by-step explanation:
Understanding Elastic and Kinetic Energy
Let's go through each part of the problem step-by-step.
A. Elastic Potential Energy (EPE) of the Spring
The Elastic Potential Energy (EPE) stored in a compressed spring can be calculated using the formula EPE = (1/2)kx^2, where k is the spring constant and x is the compression distance. In this case, k = 300 N/m and x = 30 cm (0.30 m). Plugging in the values, we get EPE = (1/2) * 300 N/m * (0.30 m)^2 = 13.5 J (Joules).
B. Kinetic Energy (KE) of the Box
When the spring returns to its natural length, all of the EPE is converted into the box's Kinetic Energy (KE). Thus, the KE of the box will also be 13.5 J, as energy is conserved in this frictionless system.
C. Velocity of the Box
To find the velocity of the box after the spring releases it, we use the formula KE = (1/2)mv^2, where m is the mass of the box and v is the velocity. We can solve for v by rearranging the formula: v = sqrt((2 * KE) / m). With m = 20 kg and KE = 13.5 J, the velocity v = sqrt((2 * 13.5 J) / 20 kg) = sqrt((27 J) / 20 kg) = sqrt(1.35) m/s ≈ 1.16 m/s. So, the box will be moving at approximately 1.16 m/s after the spring releases it.