If `E` = energy , `G`= gravitational constant, `I`=impulse and `M`=mass, then dimensions of `(GIM^(2))/(E^(2)` are same as that of

Options

A. time
B. mass
C. length
D. force

Answer

To solve the problem of finding the dimensions of \((GIM^{2})/(E^{2})\) and comparing them with the given options, we will follow these steps:

### Step 1: Find the dimensional formula for each variable

1. **Gravitational Constant (G)**:
- From the formula for gravitational force:
\[ F = \frac{G m^2}{r^2} \]
- Rearranging gives:
\[ G = \frac{F r^2}{m^2} \]
- The dimensional formula for force \(F\) is \(MLT^{-2}\) and for distance \(r\) is \(L\). Therefore:
\[ G = \frac{(MLT^{-2}) (L^2)}{M^2} = \frac{ML^3T^{-2}}{M^2} = M^{-1}L^3T^{-2} \]

2. **Impulse (I)**:
- Impulse is defined as:
\[ I = F \cdot t \]
- Using the dimensional formula for force:
\[ I = (MLT^{-2}) \cdot (T) = MLT^{-1} \]

3. **Mass (M)**:
- The dimensional formula for mass is simply:
\[ [M] = M \]

4. **Energy (E)**:
- The dimensional formula for energy is:
\[ E = \frac{1}{2}mv^2 \]
- Since \(v\) (velocity) has dimensions \(LT^{-1}\):
\[ E = M(LT^{-1})^2 = ML^2T^{-2} \]
- Therefore, the dimensional formula for \(E^2\) is:
\[ E^2 = (ML^2T^{-2})^2 = M^2L^4T^{-4} \]

### Step 2: Substitute the dimensional formulas into the expression

Now we substitute the dimensional formulas into the expression \(\frac{GIM^{2}}{E^{2}}\):

\[ \frac{GIM^{2}}{E^{2}} = \frac{(M^{-1}L^3T^{-2})(MLT^{-1})(M^2)}{M^2L^4T^{-4}} \]

### Step 3: Simplify the expression

1. **Numerator**:
- Combine the dimensions:
\[ GIM^{2} = (M^{-1}L^3T^{-2})(MLT^{-1})(M^2) = M^{-1+1+2}L^{3+1}T^{-2-1} = M^{2}L^{4}T^{-3} \]

2. **Denominator**:
- The denominator is:
\[ E^{2} = M^{2}L^{4}T^{-4} \]

3. **Final Calculation**:
- Now we have:
\[ \frac{GIM^{2}}{E^{2}} = \frac{M^{2}L^{4}T^{-3}}{M^{2}L^{4}T^{-4}} = M^{2-2}L^{4-4}T^{-3+4} = M^{0}L^{0}T^{1} = T \]

### Conclusion

The dimensions of \(\frac{GIM^{2}}{E^{2}}\) are the same as that of time \(T\).

### Final Answer
The dimensions of \((GIM^{2})/(E^{2})\) are the same as that of **Time**.

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