A ball with a mass of #160 g# is projected vertically by a spring loaded contraption. The spring in the contraption has a spring constant of #24 (kg)/s^2# and was compressed by #5/4 m# when the ball was released. How high will the ball go?

Here by the law of conservation of mechanical energy, the final potential energy of the ball at its maximum heightwill be equal to the potential energy of the compressed spring.

To find the maximum height the ball will reach, we can use the conservation of mechanical energy principle. The potential energy stored in the spring when compressed is equal to the kinetic energy of the ball when it reaches its maximum height. The potential energy stored in the spring can be calculated using the formula (PE = \frac{1}{2}kx^2), where (k) is the spring constant and (x) is the compression distance. The kinetic energy of the ball at its maximum height is equal to its potential energy, which can be expressed as (KE = mgh), where (m) is the mass of the ball, (g) is the acceleration due to gravity, and (h) is the maximum height reached by the ball. Setting these two energies equal to each other, we have (PE = KE). Solving for (h), we get (h = \frac{kx^2}{2mg}). Substituting the given values (k = 24 , \text{(kg)/s}^2), (x = \frac{5}{4} , \text{m}), (m = 0.160 , \text{kg}), and (g = 9.8 , \text{m/s}^2), we can calculate the maximum height. Thus, (h = \frac{24 \times \left(\frac{5}{4}\right)^2}{2 \times 0.160 \times 9.8}). Calculating this expression yields the maximum height the ball will reach.