Explanation:
When a projectile is launched at an angle, it follows a curved trajectory due to the influence of gravity. The range of a projectile refers to the horizontal distance covered by the projectile before hitting the ground. In this scenario, we have two different angles of projection that result in the same range.

Time of Flight:
The time of flight of a projectile refers to the total duration it remains in the air before hitting the ground. In this case, we have two different angles of projection, resulting in different times of flight: t1 and t2.

Range:
The range of a projectile can be determined using the formula: R = (v^2 * sin(2θ)) / g, where R represents the range, v is the velocity of projection, θ is the angle of projection, and g is the acceleration due to gravity.

Equal Range:
If two different angles of projection result in the same range, it implies that the two angles have the same horizontal distance covered. Mathematically, we can express this as: (v1^2 * sin(2θ1)) / g = (v2^2 * sin(2θ2)) / g.

Equating the Ranges:
By equating the ranges, we can eliminate the gravitational acceleration term and solve for the ratio of the velocities of projection. Rearranging the equation, we get: (v1^2 * sin(2θ1)) = (v2^2 * sin(2θ2)).

Ratio of Velocities:
Taking the square root of both sides of the equation, we obtain: v1 * sqrt(sin(2θ1)) = v2 * sqrt(sin(2θ2)). Dividing both sides by sqrt(sin(2θ2)), we get: v1 / v2 = sqrt(sin(2θ2)) / sqrt(sin(2θ1)).

Conclusion:
Therefore, for two angles of projection resulting in the same range, the ratio of the velocities of projection can be determined using the equation: v1 / v2 = sqrt(sin(2θ2)) / sqrt(sin(2θ1)). This equation allows us to find the velocity of projection for the given scenario.