Two projectiles `A` and `B` are thrown with initial velocities of `40 m/s` and `60 m/s` at angles `30^@` and `60^@` with the horizontal respectively. The ratio of their ranges respectively is `(g=10m/s^(2)`

Options

A. `2:sqrt3`
B. `sqrt3:2`
C. `4:9`
D. `1:1`

Answer

To find the ratio of the ranges of two projectiles A and B thrown with initial velocities of 40 m/s and 60 m/s at angles of 30° and 60° respectively, we can follow these steps:

### Step 1: Determine the Range Formula
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,
- \( g \) is the acceleration due to gravity (10 m/s² in this case).

### Step 2: Calculate the Range of Projectile A
For projectile A:
- Initial velocity \( u_A = 40 \, \text{m/s} \)
- Angle \( \theta_A = 30^\circ \)

Using the range formula:
\[ R_A = \frac{u_A^2 \sin(2\theta_A)}{g} \]
Calculating \( \sin(2\theta_A) \):
\[ \sin(2 \times 30^\circ) = \sin(60^\circ) = \frac{\sqrt{3}}{2} \]
Now substituting the values:
\[ R_A = \frac{(40)^2 \cdot \frac{\sqrt{3}}{2}}{10} \]
\[ R_A = \frac{1600 \cdot \frac{\sqrt{3}}{2}}{10} = \frac{1600\sqrt{3}}{20} = 80\sqrt{3} \, \text{m} \]

### Step 3: Calculate the Range of Projectile B
For projectile B:
- Initial velocity \( u_B = 60 \, \text{m/s} \)
- Angle \( \theta_B = 60^\circ \)

Using the range formula:
\[ R_B = \frac{u_B^2 \sin(2\theta_B)}{g} \]
Calculating \( \sin(2\theta_B) \):
\[ \sin(2 \times 60^\circ) = \sin(120^\circ) = \sin(90^\circ + 30^\circ) = \cos(30^\circ) = \frac{\sqrt{3}}{2} \]
Now substituting the values:
\[ R_B = \frac{(60)^2 \cdot \frac{\sqrt{3}}{2}}{10} \]
\[ R_B = \frac{3600 \cdot \frac{\sqrt{3}}{2}}{10} = \frac{3600\sqrt{3}}{20} = 180\sqrt{3} \, \text{m} \]

### Step 4: Calculate the Ratio of the Ranges
Now we find the ratio of the ranges \( \frac{R_A}{R_B} \):
\[ \frac{R_A}{R_B} = \frac{80\sqrt{3}}{180\sqrt{3}} = \frac{80}{180} = \frac{4}{9} \]

### Final Answer
The ratio of the ranges of projectiles A and B is:
\[ \frac{R_A}{R_B} = \frac{4}{9} \]

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