If T is the total time of flight, h is the maximum height and R is the range for horizontal motion, the x and y coordinates of projectile motion and time t are related as

Options

A. `y = 4h (t/T)(1-t/T)`
B. `y = 4h (x/R)(1-x/R)`
C. `y = 4h (T/t)(1-T/t)`
D. `y = 4h (R/x)(1-R/x)`

Explanation

To derive the relationship between the x and y coordinates of projectile motion in terms of time \( t \), total time of flight \( T \), maximum height \( h \), and range \( R \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Basic Equations**:
- The total time of flight \( T \) for a projectile is given by:
\[ T = \frac{2u \sin \theta}{g} \]
- The maximum height \( h \) reached by the projectile is given by:
\[ h = \frac{u^2 \sin^2 \theta}{2g} \]
- The range \( R \) of the projectile is given by:
\[ R = u \cos \theta \cdot T \]

2. **Expressing x and y Coordinates**:
- The x-coordinate at any time \( t \) can be expressed as:
\[ x = u \cos \theta \cdot t \]
- The y-coordinate at any time \( t \) can be expressed as:
\[ y = u \sin \theta \cdot t - \frac{1}{2} g t^2 \]

3. **Substituting Time of Flight**:
- From the equation for \( x \), we can express \( t \) in terms of \( x \):
\[ t = \frac{x}{u \cos \theta} \]

4. **Substituting for y**:
- Substitute \( t \) into the equation for \( y \):
\[ y = u \sin \theta \left(\frac{x}{u \cos \theta}\right) - \frac{1}{2} g \left(\frac{x}{u \cos \theta}\right)^2 \]
- Simplifying this gives:
\[ y = \frac{x \tan \theta}{u \cos \theta} - \frac{g x^2}{2 u^2 \cos^2 \theta} \]

5. **Using the Maximum Height**:
- We can express \( \tan \theta \) in terms of \( h \) and \( R \):
\[ \tan \theta = \frac{h}{\frac{R}{2}} \quad \text{(using the relationship between height and range)} \]

6. **Final Relationship**:
- After substituting and simplifying, we arrive at the final equation:
\[ y = 4h \left(\frac{x}{R}\right) \left(1 - \frac{x}{R}\right) \]

### Final Result:
Thus, the relationship between the x and y coordinates of projectile motion is:
\[ y = 4h \left(\frac{x}{R}\right) \left(1 - \frac{x}{R}\right) \]