A toy gun a spring of force constant `k`. When changed before being triggered in the upward direction, the spring is compressed by a distance `x`. If the mass of the shot is `m`, on the being triggered it will go up to a height of

Options

A. (a) `(Kx^2)/(mg)`
B. (b) `(x^2)/(Kmg)`
C. (c) `(Kx^2)/(2mg)`
D. (d) `(K^2x^2)/(mg)`

Explanation

To solve the problem, we need to analyze the energy transformations involved when the spring is released and the mass is shot upwards. We will use the principle of conservation of energy.

### Step 1: Understand the initial conditions
- The spring is compressed by a distance \( x \).
- The force constant of the spring is \( k \).
- The mass of the shot is \( m \).
- Initially, the kinetic energy of the mass is zero as it is at rest.

**Hint:** Identify the initial potential energy stored in the spring when it is compressed.

### Step 2: Calculate the potential energy stored in the spring
The potential energy (PE) stored in the spring when it is compressed by a distance \( x \) is given by the formula:
\[ PE = \frac{1}{2} k x^2 \]

**Hint:** Remember that the potential energy in a spring is calculated using the spring constant and the compression distance.

### Step 3: Analyze the final conditions
When the spring is released, the potential energy converts into gravitational potential energy (GPE) as the mass rises to a height \( h \). At the maximum height, the kinetic energy will be zero again.

The gravitational potential energy at height \( h \) is given by:
\[ GPE = mgh \]

**Hint:** Think about how energy conservation applies when the spring releases its energy to lift the mass.

### Step 4: Apply the conservation of energy principle
According to the conservation of energy, the initial potential energy in the spring will equal the gravitational potential energy at the maximum height:
\[ \frac{1}{2} k x^2 = mgh \]

**Hint:** Set the potential energy from the spring equal to the gravitational potential energy to find the height.

### Step 5: Solve for height \( h \)
Rearranging the equation to solve for \( h \):
\[ mgh = \frac{1}{2} k x^2 \]
\[ h = \frac{k x^2}{2mg} \]

**Hint:** Isolate \( h \) on one side of the equation to express it in terms of the other variables.

### Final Answer
The height \( h \) to which the mass will rise is given by:
\[ h = \frac{k x^2}{2mg} \]

### Summary of Steps
1. Identify initial conditions (spring compressed, mass at rest).
2. Calculate the potential energy stored in the spring.
3. Analyze final conditions (mass rises to height \( h \)).
4. Apply conservation of energy to relate spring potential energy to gravitational potential energy.
5. Solve for height \( h \).