Two shells are fired from a cannon with a speed u each, at angles of `alpha` and `beta` respectively to the horizontal. The time interval between the shots is T. they collide in mid-air after time t from the first shot. Which of the following conditions must be satisfied ?

Options

A. `alpha` `gt beta`
B. `t cos alpha=(t-T)cos beta`
C. `(t-T)cos alpha=t cos beta`
D. `(usinalpha)t-1/2"gt"^2=(usinbeta)(t-T)-1/2g(t-T)^2`

Explanation

To solve the problem of two shells fired from a cannon at angles α and β, we need to analyze the conditions for their collision in mid-air. Here’s a step-by-step breakdown:

### Step 1: Understanding the Problem
Two shells are fired with the same speed \( u \) at angles \( \alpha \) and \( \beta \). The first shell is fired at time \( t = 0 \), and the second shell is fired after a time interval \( T \). The shells collide after a time \( t \) from the first shot.

### Step 2: Horizontal Motion Analysis
For the shells to collide, the horizontal distances traveled by both shells must be equal at the time of collision.

- **Horizontal distance of shell 1 (fired at angle α)**:
\[ x_1 = u \cos(\alpha) \cdot t \]

- **Horizontal distance of shell 2 (fired at angle β)**:
\[ x_2 = u \cos(\beta) \cdot (t - T) \]

Setting these distances equal gives us:
\[ u \cos(\alpha) \cdot t = u \cos(\beta) \cdot (t - T) \]

Cancelling \( u \) (since \( u \neq 0 \)):
\[ t \cos(\alpha) = (t - T) \cos(\beta) \]

### Step 3: Vertical Motion Analysis
Next, we analyze the vertical motion. The vertical distances traveled by both shells must also be equal at the time of collision.

- **Vertical distance of shell 1**:
\[ y_1 = u \sin(\alpha) \cdot t - \frac{1}{2} g t^2 \]

- **Vertical distance of shell 2**:
\[ y_2 = u \sin(\beta) \cdot (t - T) - \frac{1}{2} g (t - T)^2 \]

Setting these distances equal gives us:
\[ u \sin(\alpha) \cdot t - \frac{1}{2} g t^2 = u \sin(\beta) \cdot (t - T) - \frac{1}{2} g (t - T)^2 \]

### Step 4: Analyzing the Conditions
From the horizontal motion equation, we derived:
\[ t \cos(\alpha) = (t - T) \cos(\beta) \]

From the vertical motion equation, we can simplify and analyze the conditions further.

### Step 5: Conclusion
Based on the analysis, we can conclude the following conditions must be satisfied for the shells to collide:

1. **Horizontal Distance Condition**:
\[ t \cos(\alpha) = (t - T) \cos(\beta) \]

2. **Vertical Distance Condition**:
\[ u \sin(\alpha) \cdot t - \frac{1}{2} g t^2 = u \sin(\beta) \cdot (t - T) - \frac{1}{2} g (t - T)^2 \]

3. **Angle Condition**:
Since \( \cos(\alpha) < \cos(\beta) \) implies \( \alpha > \beta \) (because cosine is a decreasing function in the first quadrant), we conclude that:
\[ \alpha > \beta \]

### Summary of Conditions:
- \( t \cos(\alpha) = (t - T) \cos(\beta) \)
- \( u \sin(\alpha) \cdot t - \frac{1}{2} g t^2 = u \sin(\beta) \cdot (t - T) - \frac{1}{2} g (t - T)^2 \)
- \( \alpha > \beta \)