Question

A ball of mass 4 kg , moving with a velocity of 10 m s−1 , collides with a spring of length 8 m and force constant 100 N m−1 . The length of the compressed spring is x m . The value of x , to the nearest integer, is ___.

Answer 1

6

Answer 2

The problem involves the conversion of kinetic energy of the ball into elastic potential energy of the spring. When the ball collides with the spring, its kinetic energy is used to compress the spring. At the point of maximum compression, the ball momentarily comes to rest, and all its initial kinetic energy is stored as potential energy in the spring.

1. Calculate the initial kinetic energy of the ball: The mass of the ball ($m$) = 4 kg The velocity of the ball ($v$) = 10 m/s Kinetic Energy ($KE$) is given by the formula: $KE = \frac{1}{2}mv^2$ $KE = \frac{1}{2} \times 4 \text{ kg} \times (10 \text{ m/s})^2 = 200 \text{ J}$

2. Equate kinetic energy to the potential energy stored in the spring: The force constant of the spring ($k$) = 100 N/m Let the compression of the spring be $x_{compression}$. The Potential Energy ($PE$) stored in the spring is given by the formula: $PE = \frac{1}{2}kx_{compression}^2$ According to the principle of conservation of energy (assuming no energy loss due to friction or sound), the initial kinetic energy of the ball is completely converted into the potential energy of the spring: $KE = PE$ $200 \text{ J} = \frac{1}{2} \times 100 \text{ N/m} \times x_{compression}^2$

3. Solve for the compression ($x_{compression}$): $200 = 50 \times x_{compression}^2$ $x_{compression}^2 = \frac{200}{50} = 4$ $x_{compression} = \sqrt{4} = 2 \text{ m}$ (Since compression is a physical length, we take the positive root)

4. Calculate the final length of the compressed spring (x): The natural length of the spring ($L_{natural}$) = 8 m The compressed length ($x$) is the natural length minus the compression: $x = L_{natural} - x_{compression}$ $x = 8 \text{ m} - 2 \text{ m} = 6 \text{ m}$

The value of x, to the nearest integer, is 6.