To solve the problem, we need to determine the force \( F \) applied to the catapult that launches a stone of mass 500 g (0.5 kg) with a velocity of 40 m/s after being extended by 20 cm (0.2 m). We will use the principles of energy conservation and Newton's second law to find the correct answer. ### Step-by-Step Explanation 1. **Convert Mass to Kilograms**: The mass of the stone is given as 500 g. We need to convert this to kilograms because the standard unit of mass in physics is kilograms (kg). \[ m = 500 \, \text{g} = 0.5 \, \text{kg} \] 2. **Calculate the Kinetic Energy of the Stone**: When the stone is launched, it has kinetic energy given by the formula: \[ KE = \frac{1}{2} mv^2 \] where \( m \) is the mass and \( v \) is the velocity. Plugging in the values: \[ KE = \frac{1}{2} \times 0.5 \, \text{kg} \times (40 \, \text{m/s})^2 \] \[ KE = \frac{1}{2} \times 0.5 \times 1600 = 0.25 \times 1600 = 400 \, \text{J} \] 3. **Calculate the Work Done by the Force**: The work done on the stone by the force \( F \) when the catapult is extended is given by: \[ W = F \cdot d \] where \( d \) is the distance over which the force is applied (0.2 m). Since the work done on the stone is equal to the kinetic energy it gains, we can set these equal: \[ F \cdot 0.2 \, \text{m} = 400 \, \text{J} \] 4. **Solve for the Force \( F \)**: Rearranging the equation to solve for \( F \): \[ F = \frac{400 \, \text{J}}{0.2 \, \text{m}} = 2000 \, \text{N} \] 5. **Final Answer**: The calculated force \( F \) is 2000 N, which can be expressed in scientific notation as: \[ F = 2.0 \times 10^3 \, \text{N} \] ### Explanation of Options - **Option A: \( 4.0 \times 10^2 \, \text{N} \)**: This value is too low. It would imply that the work done was only 80 J, which is insufficient to give the stone the kinetic energy of 400 J. - **Option B: \( 2.0 \times 10^3 \, \text{N} \)**: This is the correct answer. It matches our calculated force based on the work-energy principle. - **Option C: \( 4.0 \times 10^3 \, \text{N} \)**: This value is too high. It would imply that the work done was 800 J, which exceeds the kinetic energy of the stone. - **Option D: \( 4.0 \times 10^4 \, \text{N} \)**: This value is excessively high and would imply an unrealistically large amount of work done (8000 J), far beyond what is needed to achieve the stone's kinetic energy. ### Common Pitfalls - **Forgetting to Convert Units**: Always ensure that mass is in kilograms and distance in meters when using SI units. - **Misapplying the Work-Energy Principle**: Remember that the work done on an object is equal to the change in kinetic energy of that object. - **Neglecting the Direction of Force**: In this case, the force is applied in the direction of the displacement, which is crucial for calculating work. ### Revision Summary - The mass of the stone must be converted to kilograms for calculations. - Kinetic energy is calculated using \( KE = \frac{1}{2} mv^2 \). - Work done by the force is equal to the kinetic energy gained by the stone. - The correct force \( F \) is calculated using \( F = \frac{KE}{d} \). The correct answer is **B: \( 2.0 \times 10^3 \, \text{N} \)**.