At what angle with the horizontal should a ball be thrown so that the range `R` is related to the time of flight as `R = 5 T^2` ? `(Take g = 10 ms6-2)`.

Options

A. `30^@`
B. `45^@`
C. `60^@`
D. `90^@`

Explanation

To find the angle at which a ball should be thrown so that the range \( R \) is related to the time of flight \( T \) by the equation \( R = 5T^2 \), we can follow these steps:

### Step 1: Write the equations for range and time of flight
The range \( R \) of a projectile is given by the formula:
\[ R = \frac{u^2 \sin 2\theta}{g} \]
where \( u \) is the initial velocity, \( \theta \) is the angle of projection, and \( g \) is the acceleration due to gravity.

The time of flight \( T \) is given by:
\[ T = \frac{2u \sin \theta}{g} \]

### Step 2: Substitute \( T \) into the range equation
From the equation for time of flight, we can express \( u \sin \theta \) in terms of \( T \):
\[ u \sin \theta = \frac{gT}{2} \]
Now, substituting this into the range equation, we get:
\[ R = \frac{u^2 \sin 2\theta}{g} = \frac{u^2 (2 \sin \theta \cos \theta)}{g} \]
Substituting \( u \sin \theta = \frac{gT}{2} \):
\[ u^2 = \left(\frac{gT}{2 \sin \theta}\right)^2 \]
Thus, we can write:
\[ R = \frac{\left(\frac{gT}{2 \sin \theta}\right)^2 (2 \sin \theta \cos \theta)}{g} \]
This simplifies to:
\[ R = \frac{g^2 T^2}{4 \sin^2 \theta} \cdot \frac{2 \sin \theta \cos \theta}{g} \]
\[ R = \frac{g T^2 \cos \theta}{2 \sin \theta} \]

### Step 3: Set the equation \( R = 5T^2 \)
Now, we set the expression for \( R \) equal to \( 5T^2 \):
\[ \frac{g T^2 \cos \theta}{2 \sin \theta} = 5T^2 \]
Dividing both sides by \( T^2 \) (assuming \( T \neq 0 \)):
\[ \frac{g \cos \theta}{2 \sin \theta} = 5 \]

### Step 4: Solve for \( \theta \)
Rearranging gives:
\[ g \cos \theta = 10 \sin \theta \]
Dividing both sides by \( g \) (where \( g = 10 \, \text{m/s}^2 \)):
\[ \cos \theta = \sin \theta \]
This implies:
\[ \tan \theta = 1 \]
Thus, the angle \( \theta \) is:
\[ \theta = 45^\circ \]

### Conclusion
The angle at which the ball should be thrown is \( 45^\circ \).

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