Determining the Range of a Projectile Launched from a Height

Understanding how to calculate the range of a projectile launched from a specific height, such as 1 meter, is crucial in various fields, including engineering, physics, and ballistics. This article explains step-by-step how to determine the range of a projectile given an initial velocity and an initial height. We'll delve into the physics behind the calculations, providing clear explanations and examples.

Initial Conditions and Key Variables

Let's start by understanding the initial conditions and the key variables involved in calculating the range of a projectile:

Height of launch (h): 1 meter Initial velocity (v): 5 m/s Initial angle of launch (θ): This determines both the vertical and horizontal components of the velocity Gravitational acceleration (g): 9.8 m/s2

The vertical component (vv) of the initial velocity (v) is given by vv v sinθ, and the horizontal component (vh) is given by vh v cosθ.

Calculating the Time of Flight

First, we need to calculate the time of flight for the projectile. The time of flight consists of the time the projectile is ascending (tup), the time it remains at its highest point (th), and the time it falls back to the ground (tdn).

The time to reach the maximum height (tup) can be calculated using the vertical component of the velocity:

tup vv / g v sinθ / g

Since the time ascending is equal to the time descending (tdn tup), the total time of flight (tflight) is simply twice the time to reach the maximum height:

tflight 2 tup 2 (v sinθ / g)

Additionally, we need to account for the time it takes for the projectile to fall from the initial height (h) to the ground:

tdrop from h √(2h / g)

Calculating the Total Height Fallen

The total height fallen by the projectile after reaching the highest point can be calculated using the horizontal distance (dh) covered during the time it falls:

dh vhtflight v cosθ (2vv / g) v cosθ (2 sinθ / g) vh (2 vv / g)

Substituting v 5 m/s and g 9.8 m/s2, we can calculate the total height fallen as:

dh cosθ (2 (5 sinθ) / 9.8) cosθ (2 (5 sinθ) / 9.8) cosθ (10 sinθ / 9.8)

For example, if the angle of launch is 25 degrees:

cos25 0.9063

sin25 0.4226

dh 0.9063 (10 * 0.4226 / 9.8) 0.9063 * (4.226 / 9.8) 0.9063 * 0.4313 ≈ 0.392 meters

Final Steps to Calculate the Range

With the information from the previous steps, we can now calculate the range of the projectile. The range (R) is the horizontal distance covered during the total time of flight:

R vh * tflight v cosθ * (2 vv / g)

Substituting the values for initial velocity (v 5 m/s), gravitational acceleration (g 9.8 m/s2), and the trigonometric components, we get:

R 5 cos25 * (2 * 5 sin25 / 9.8) 5 * 0.9063 * (2 * 5 * 0.4226 / 9.8) 5 * 0.9063 * (4.226 / 9.8) 5 * 0.9063 * 0.4313 ≈ 1.91 meters

Conclusion

By understanding the key variables and applying the correct formulas from kinematics, one can accurately determine the range of a projectile launched from a height. The process involves breaking down the initial velocity into its vertical and horizontal components and then calculating the time of flight and the total horizontal distance covered.

Summary of Key Formulas

Vertical component of velocity: vv v sinθ Horizontal component of velocity: vh v cosθ Time to reach maximum height: tup vv / g Total time of flight: tflight 2 tup Total height fallen: tdrop from h √(2h / g) Range: R vh * tflight v cosθ * (2 vv / g)