Projectile Motion and Air Resistance: Calculating the Final Speed of a Stone Thrown from a Building

The problem at hand is to calculate the speed of a stone just before it strikes the ground when thrown from the top of a building upward at an angle of 30 degrees to the horizontal with an initial speed of 20.0 m/s. The height of the building is 45.0 m. This involves the calculation of velocity components, time of flight, and the effect of air resistance.

Initial Velocity Components

The initial velocity of the stone is decomposed into horizontal and vertical components:

Horizontal component: v_x v_0 cdot cos(30°) approx 17.32 text{ m/s}

Vertical component: v_{0y} v_0 cdot sin(30°) 10.0 text{ m/s}

Time of Flight

Using the vertical motion equation, we can determine the time it takes for the stone to reach the ground:

y  v_{0y} cdot t - frac{1}{2} g t^2

Given that the vertical displacement is -45.0 m, we reformulate the equation:

-45.0  10.0 t - frac{1}{2} cdot 9.81 t^2

This equation can be simplified into a standard quadratic equation:

4.905 t^2 - 10.0 t - 45.0  0

Solving this quadratic equation yields the time of flight:

t  frac{-b pm sqrt{b^2 - 4ac}}{2a}

where a 4.905, b -10.0, and c -45.0. The discriminant is calculated and two possible values of t are found. The physically meaningful solution is:

t approx 4.21 text{ s}

Final Velocity Calculation

The final vertical velocity just before the stone hits the ground is:

v_{fy}  v_{0y} - g cdot t  10.0 - 9.81 cdot 4.21 approx -31.30 text{ m/s}

The horizontal velocity remains constant throughout, so:

v_{fx}  v_x approx 17.32 text{ m/s}

The final speed can be calculated using the Pythagorean theorem:

v_f  sqrt{v_{fx}^2   v_{fy}^2}  sqrt{17.32^2   (-31.30)^2} approx 35.8 text{ m/s}

Impact of Air Resistance

While the above calculations ignore air resistance, in reality, it significantly affects the motion of the stone. The lighter the object, the more it is slowed down due to air resistance. For instance, a ping pong ball decelerates from 20 m/s to 6.5 m/s and accelerates back to 9 m/s, while a 42-pound cannon ball decelerates to 17 m/s and accelerates back to 35 m/s.

Due to air resistance, the stone's trajectory would differ, and its final speed may be lower than the calculated 35.8 m/s. The trajectories of these balls would reveal the effects of air resistance, with heavier objects experiencing less deceleration.

Conclusion

The speed of the stone just before it strikes the ground is approximately 35.8 m/s, considering the initial velocity, height of the building, and the principles of projectile motion. Real-world calculations would need to account for air resistance, which varies based on the mass and shape of the object.