Calculating Launch Angles for Projectiles with Same Time of Flight

About the Problem

Imagine two projectiles launched with different initial velocities but having the same time of flight. Specifically, we have a projectile launched at 30 m/s and another at 60 m/s. Both have a time of flight of 6 seconds. The question seeks to determine the angles of projection for each projectile to achieve this time of flight.

Understanding Projectile Motion

In projectile motion, the path of a projectile is influenced by gravity. The vertical motion of the projectile is purely affected by gravity, regardless of its horizontal velocity. The key formula that helps us in this problem is the time of flight, which is given by:

Time of Flight (frac{2u sin theta}{g}) Time of Flight (T): 6 seconds. Initial Velocity (u): 30 m/s and 60 m/s for the two projectiles. Acceleration due to Gravity (g): 9.8 m/s2. Sine of Projection Angle ((sin theta)): This determines the vertical component of the initial velocity.

Mathematical Solution

The steps to solve for the launch angles are as follows:

For the First Projectile (30 m/s)

Set the time of flight formula for the first projectile: T1 (frac{2 cdot 30 cdot sin theta_1}{9.8}) Given T1 6 seconds, solve for (sin theta_1): 6 (frac{60 cdot sin theta_1}{9.8}) (sin theta_1 frac{6 cdot 9.8}{60} 0.98) (theta_1 arcsin(0.98) approx 78.81^circ)

For the Second Projectile (60 m/s)

Set the time of flight formula for the second projectile: T2 (frac{2 cdot 60 cdot sin theta_2}{9.8}) Given T2 6 seconds, solve for (sin theta_2): 6 (frac{120 cdot sin theta_2}{9.8}) (sin theta_2 frac{6 cdot 9.8}{120} 0.49) (theta_2 arcsin(0.49) approx 29.37^circ)

The angles of projection for the two projectiles are approximately 78.81° and 29.37°.

Explanation of the Results

The larger initial velocity (60 m/s) results in a smaller launch angle (29.37°), while the smaller initial velocity (30 m/s) results in a larger launch angle (78.81°). This is due to the relationship between the sine of the angle and the initial velocity. The sine of the angle must remain constant for the time of flight to be the same, but a higher initial velocity requires a smaller angle to achieve the same flight time due to the vertical component of the velocity.

Conclusion

In conclusion, understanding projectile motion and the impact of initial velocities on the launch angle is crucial for solving such problems. The two projectiles in this scenario, with different initial velocities but the same time of flight, indeed have distinct launch angles. By applying the time of flight formula, we can determine these angles accurately.