Question

A ball of mass 4 kg, moving with a velocity of 10 ms\(^{-1}\), collides with a spring of length 8 m and force constant 100 Nm\(^{-1}\). The length of the compressed spring is x m. The value of x, to the nearest integer, is __________.

Hint:

Always distinguish between the compression (\(\Delta L\)) and the final length (\(L - \Delta L\)).

Answer 1

Correct Answer: 6

Step 1: Use Conservation of Mechanical Energy. The kinetic energy of the ball is converted into elastic potential energy of the spring at maximum compression. \[ \frac{1}{2} m v^2 = \frac{1}{2} k (\Delta L)^2 \]
Step 2: Solve for compression (\(\Delta L\)). \[ 4 \times (10)^2 = 100 \times (\Delta L)^2 \implies 400 = 100 (\Delta L)^2 \] \[ (\Delta L)^2 = 4 \implies \Delta L = 2 \text{ m} \]
Step 3: Find the final length of the spring (\(x\)). \[ x = L_{initial} - \Delta L = 8 \text{ m} - 2 \text{ m} = 6 \text{ m} \]