A Real-World Example of Projectile Motion

Imagine you are a curious student looking to understand linear motion in a practical scenario. A projectile is fired horizontally from a gun that is 45.0 meters above flat ground, emerging with a speed of 250 meters per second. The question at hand is: how long does the projectile remain in the air?

The Myth Busting Approach

Now, if I were to simply dismiss this as an exercise in academic dishonesty, you might feel less inclined to read on. The first thing to realize is that the horizontal velocity of the projectile does not affect its vertical motion. The projectile will fall due to gravity at the same rate as if it were dropped from the same height.

Gravitational Acceleration and the Problem

The key element here is gravitational acceleration. On Earth, the gravitational acceleration is approximately 9.81 m/s2. However, for simplicity, we will approximate this to 10 m/s2. The difference in the answer between using 9.81 and 10 is only 0.03047662390 seconds, which is generally negligible in most practical applications.

The Standard Equation for Free Fall

The standard equation for free fall motion is given by 1/2 * a * t2 u * t - s 0, where s is the distance, a is the acceleration, t is time, and u is the initial velocity. In this case, the distance s is 45 meters, the acceleration a is 10 m/s2, and the initial velocity u is 0 m/s.

Deriving the Solution

We start with the equation and plug in our values:

1/2 * 10 * t2 0 * t - 45 0

Simplifying, we get:

5 * t2 - 45 0

Further simplification gives us:

t2 9

And taking the square root of both sides:

t 3.16227766016 seconds

Real-life Considerations

While we could delve into more complex scenarios, such as the curvature of the Earth, the Magnus Effect, or atmospheric drag, these factors are generally negligible for a height of 45 meters and a muzzle velocity of 250 m/s. These effects become significantly more important at much larger scales, such as in orbital mechanics or hypersonic travel.

Conclusion

Thus, the projectile remains in the air for approximately 3.16 seconds. This problem demonstrates the application of classical physics principles in a practical context, shedding light on how vertical motion is independent of horizontal velocity.