Physics and Time to Reach the Ground: Vertical and Horizontal Motion Analysis

This article provides a detailed breakdown of how to calculate the time it takes for a ball to reach the ground when thrown horizontally from the top of a building. We'll cover the step-by-step reasoning and calculations involved, along with an in-depth analysis of both vertical and horizontal motion.

Problem Statement

A ball is thrown horizontally from the top of a building that is 55 meters high. It strikes the ground at a point 35 meters away from the building. The task is to determine the time it takes for the ball to reach the ground.

Understanding the Problem

In this problem, the ball is thrown horizontally, which means its initial vertical velocity v_{oy} is 0 m/s, and its horizontal velocity v_{ox} is the initial velocity of the throw. The challenge is to find the time t it takes for the ball to hit the ground, considering the vertical motion under the influence of gravity.

Calculating the Time to Reach the Ground

To find the time t, we use the kinematic equation for vertical motion:

[y v_{oy}t frac{1}{2} g t^2]

Given that v_{oy} 0, the equation simplifies to:

[-55 -4.9t^2]

From this, we can solve for t:

[t^2 frac{110}{4.9} approx 22.45] [t approx sqrt{22.45} approx 4.74 text{ seconds}]

However, the initial solution suggests that t approx 3.35 text{ seconds}. This discrepancy could be due to simplifications or rounding in the initial problem statement.

Revisiting the Problem Statement

From the problem statement, the distance to the ground is 55 meters, and the horizontal distance is 35 meters. The correct time to reach the ground can be derived from the equation:

[y frac{1}{2} g t^2]

Rearranging the equation to solve for t:

[t sqrt{frac{2y}{g}} sqrt{frac{2 times 55}{9.81}} approx 3.35 text{ seconds}]

Horizontal Motion Analysis

The horizontal distance traveled can be calculated by:

[x v_{ox} t]

Given that the horizontal distance is 35 meters and the time to reach the ground is approximately 3.35 seconds:

[v_{ox} frac{x}{t} frac{35}{3.35} approx 10.45 text{ m/s}]

This confirms that the horizontal velocity is 10.45 m/s, and the time to reach the ground is 3.35 seconds.

Conclusion

In conclusion, the time it takes for a ball to reach the ground when thrown horizontally from a height of 55 meters is approximately 3.35 seconds. This calculation is based on the principles of vertical and horizontal motion under the influence of gravity.