Question

Two projectiles u₁ and u₂, with speed of 40 m/s and 60 m/s respectively are thrown at a certain angle, u₁ with 30^o and u₂ with 60^o, what is the ratio of range?

Answer 1

Correct Answer: 49

The range of a projectile is given by: 

\[ R=\frac{u^2\sin 2\theta}{g} \] 
Step 1: Find range of first projectile

For the first projectile: 

\[ u_1=40\text{ m/s}, \quad \theta_1=30^\circ \] So, 

\[ R_1=\frac{40^2\sin 60^\circ}{g} \] \[ R_1=\frac{1600\sin 60^\circ}{g} \] 
Step 2: Find range of second projectile

For the second projectile: 

\[ u_2=60\text{ m/s}, \quad \theta_2=60^\circ \] So, 

\[ R_2=\frac{60^2\sin 120^\circ}{g} \] \[ R_2=\frac{3600\sin 120^\circ}{g} \] 
Step 3: Use trigonometric identity

Since, 

\[ \sin 60^\circ=\sin 120^\circ=\frac{\sqrt{3}}{2} \] therefore, 

\[ R_1=\frac{1600\cdot \frac{\sqrt{3}}{2}}{g} \] \[ R_2=\frac{3600\cdot \frac{\sqrt{3}}{2}}{g} \] 
Step 4: Find the ratio

\[ \frac{R_1}{R_2}=\frac{1600}{3600}=\frac{4}{9} \] Hence, 

\[ R_1:R_2=4:9 \] 
Final Answer:

\[ \boxed{4:9} \]