A projectile fired at an angle of `45^(@)` travels a total distance R, called the range, which depends only on the initial speed v and the acceleration of gravity g. Using dimensional analysis, find how R depends on the speed and on g.

Explanation

To solve the problem using dimensional analysis, we need to find how the range \( R \) of a projectile depends on the initial speed \( v \) and the acceleration due to gravity \( g \).

### Step-by-Step Solution:

1. **Identify the dimensions of the quantities involved:**
- The range \( R \) is a distance, which has the dimensional formula:
\[ [R] = L^1 \]
- The initial speed \( v \) has the dimensional formula:
\[ [v] = L^1 T^{-1} \]
- The acceleration due to gravity \( g \) has the dimensional formula:
\[ [g] = L^1 T^{-2} \]

2. **Express the relationship between \( R \), \( v \), and \( g \):**
- We assume that the range \( R \) depends on \( v \) and \( g \) in the following way:
\[ R \propto v^a \cdot g^b \]
- Therefore, we can write:
\[ [R] = [v]^a \cdot [g]^b \]
- Substituting the dimensional formulas, we have:
\[ L^1 = (L^1 T^{-1})^a \cdot (L^1 T^{-2})^b \]

3. **Expand the right-hand side:**
- This gives us:
\[ L^1 = L^{a+b} \cdot T^{-a-2b} \]

4. **Set up equations by comparing dimensions:**
- From the equation \( L^1 = L^{a+b} \), we can equate the powers of \( L \):
\[ a + b = 1 \quad \text{(Equation 1)} \]
- From the equation \( T^0 = T^{-a-2b} \), we can equate the powers of \( T \):
\[ -a - 2b = 0 \quad \text{(Equation 2)} \]

5. **Solve the system of equations:**
- From Equation 2, we can express \( a \) in terms of \( b \):
\[ a = -2b \]
- Substitute \( a = -2b \) into Equation 1:
\[ -2b + b = 1 \implies -b = 1 \implies b = -1 \]
- Now substitute \( b = -1 \) back into the expression for \( a \):
\[ a = -2(-1) = 2 \]

6. **Write the final relationship:**
- Substituting the values of \( a \) and \( b \) back into the proportional relationship gives:
\[ R \propto v^2 \cdot g^{-1} \]
- This can be expressed as:
\[ R = k \frac{v^2}{g} \]
- where \( k \) is a constant of proportionality.

### Conclusion:
The range \( R \) of a projectile fired at an angle of \( 45^\circ \) depends on the initial speed \( v \) and the acceleration due to gravity \( g \) as follows:
\[ R \propto \frac{v^2}{g} \]