Time to Hit the Ground: Comparing Dropped and Horizontally Thrown Rocks

When two rocks are placed at the top of a building, one dropped from rest and the other thrown horizontally, which one will hit the ground first? This question dives into the principles of physics and helps us understand how gravity influences the motion of objects in different scenarios.

Understanding the Initial Conditions

Consider a scenario where we have Rock 1 and Rock 2 placed at the top of a building. Rock 1 is released from rest, meaning it starts its descent purely under the influence of gravity. In contrast, Rock 2 is given an initial horizontal velocity of 5 m/s before it is released.

Mathematical Analysis for Both Rocks

To determine the time taken for each rock to hit the ground, we can use the basic kinematic equations of motion. Let's analyze the vertical motion of both rocks separately.

Rock 1: Dropped from Rest

For Rock 1:

Initial vertical velocity ( u 0 ) m/s Vertical acceleration ( a 9.81 , text{m/s}^2 ) (acceleration due to gravity) Vertical displacement ( s h ) (the height of the building)

Using the second equation of motion:

$$ s ut frac{1}{2} a t^2 $$

Substituting the values:

$$ h 0 cdot t frac{1}{2} cdot 9.81 t^2 $$ $$ h frac{1}{2} cdot 9.81 t^2 $$

Solving for ( t ):

$$ t^2 frac{2h}{9.81} $$ $$ t sqrt{frac{2h}{9.81}} $$

Rock 2: Thrown Horizontally

For Rock 2:

Initial vertical velocity ( u 0 ) m/s (the vertical component is still zero) Vertical acceleration ( a 9.81 , text{m/s}^2 ) Vertical displacement ( s h ) (the height of the building)

The analysis for the vertical motion of Rock 2 is identical to Rock 1:

$$ h 0 cdot t frac{1}{2} cdot 9.81 t^2 $$ $$ h frac{1}{2} cdot 9.81 t^2 $$

Solving for ( t ):

$$ t^2 frac{2h}{9.81} $$ $$ t sqrt{frac{2h}{9.81}} $$

Conclusion: Time to Hit the Ground

Both rocks take the same amount of time to hit the ground. The time to fall is determined only by the height from which they are dropped and the acceleration due to gravity. Therefore, the time taken for both Rock 1 and Rock 2 to hit the ground is identical.

Horizontal Motion and Acceleration

Rock 2's horizontal velocity of 5 m/s is constant throughout its flight. Since there is no acceleration in the horizontal direction, this horizontal velocity does not affect the time it takes to reach the ground. The horizontal motion and the vertical motion are independent of each other.

Even in an atmosphere with negligible friction, the horizontal velocity of Rock 2 does not influence its downward motion. Both rocks experience the same gravitational acceleration and thus take the same amount of time to hit the ground.

Conclusion

In summary, the time taken for Rock 1 and Rock 2 to hit the ground is the same, regardless of their initial horizontal velocity. This is because the vertical motion of both rocks is solely influenced by gravity, and horizontal motion does not impact the time to reach the ground in the absence of air resistance.