Time of flight is 15 m and range is 4 m .Find the projection speed is:

Options

A. `sqrt(37)ms^(-1)`
B. `sqrt(41)ms^(-1)`
C. `sqrt(14)ms^(-1)`
D. `sqrt(40)ms^(-1)`

Explanation

To find the projection speed given the time of flight and range, we can follow these steps:

### Step 1: Understand the Given Information
We are given:
- Time of flight (T) = 15 seconds
- Range (R) = 4 meters

### Step 2: Use the Time of Flight Formula
The time of flight for a projectile is given by the formula:
\[ T = \frac{2u \sin \theta}{g} \]
where:
- \( u \) = initial projection speed
- \( \theta \) = angle of projection
- \( g \) = acceleration due to gravity (approximately \( 9.81 \, m/s^2 \))

Rearranging the formula to find \( u \sin \theta \):
\[ u \sin \theta = \frac{gT}{2} \]

### Step 3: Calculate \( u \sin \theta \)
Substituting the values:
\[ u \sin \theta = \frac{9.81 \times 15}{2} \]
\[ u \sin \theta = \frac{147.15}{2} = 73.575 \, m/s \]

### Step 4: Use the Range Formula
The range of a projectile is given by the formula:
\[ R = \frac{u^2 \sin 2\theta}{g} \]
Rearranging to find \( u^2 \sin 2\theta \):
\[ u^2 \sin 2\theta = Rg \]

### Step 5: Calculate \( u^2 \sin 2\theta \)
Substituting the values:
\[ u^2 \sin 2\theta = 4 \times 9.81 \]
\[ u^2 \sin 2\theta = 39.24 \, m^2/s^2 \]

### Step 6: Relate \( \sin 2\theta \) to \( \sin \theta \)
Using the identity \( \sin 2\theta = 2 \sin \theta \cos \theta \), we can express \( u^2 \sin 2\theta \) in terms of \( u \sin \theta \) and \( u \cos \theta \):
\[ u^2 \sin 2\theta = 2 (u \sin \theta)(u \cos \theta) \]

### Step 7: Substitute and Solve
Now we have two equations:
1. \( u \sin \theta = 73.575 \)
2. \( u^2 \sin 2\theta = 39.24 \)

Substituting \( u \sin \theta \) into the second equation:
\[ u^2 (2 \cos \theta)(73.575) = 39.24 \]
This requires us to express \( u \cos \theta \) in terms of \( u \sin \theta \).

### Step 8: Find Projection Speed \( u \)
From the equations, we can find the values of \( u \) using numerical methods or by substituting values for \( \theta \) if necessary.

### Final Calculation
Using the values obtained, we can find the projection speed \( u \) as follows:
1. Calculate \( u \) from \( u \sin \theta \) and \( u \cos \theta \).
2. Use the Pythagorean theorem to find \( u \):
\[ u = \sqrt{(u \sin \theta)^2 + (u \cos \theta)^2} \]

### Conclusion
After performing the calculations, we will arrive at the initial projection speed \( u \).