How Long Does a Stone Take to Reach the Bottom of a Cliff When Thrown Horizontally?

A classic physics problem involves a stone being thrown horizontally from the top of a cliff. The question often asks how long it takes for the stone to reach the bottom of the cliff. This problem illustrates the concept of free fall and how it is independent of horizontal motion. In this article, we will explore the physics behind such a scenario and provide a step-by-step solution.

Understanding the Scenario and Physics Involved

When a stone is thrown horizontally from the top of a cliff, it starts moving with a certain horizontal velocity (15 m/s in this case) while simultaneously experiencing the pull of gravity. The key point to remember is that the horizontal motion does not influence the time it takes for the stone to fall vertically. The time to reach the bottom is determined solely by the height of the cliff and gravity.

Let us break down the problem using a step-by-step approach:

Step 1: Understanding the Physics

The equation to find the time it takes for an object to fall a certain height is given by:

h frac{1}{2} g t^2

Where:

h is the height (90 m in this case) g is the acceleration due to gravity (9.81 m/s2) t is the time in seconds

Step 2: Rearranging the Equation

Rearranging the equation to solve for time:

t sqrt{frac{2h}{g}}

Substituting the values:

t sqrt{frac{2 times 90 , text{m}}{9.81 , text{m/s}^2}}

Calculating this gives:

t sqrt{frac{180}{9.81}} approx sqrt{18.35} approx 4.29 , text{s}

Therefore, it would take approximately 4.29 seconds for the stone to reach the bottom of the cliff.

Step 3: Additional Insights

Horizontal motion (15 m/s in this case) has no impact on the time taken for the stone to fall vertically. The horizontal velocity affects the distance covered in the horizontal direction, not the time of fall. To determine the distance, you would need to multiply the horizontal velocity by the time of fall (4.29 seconds).

What Factors Affect the Time of Fall?

The only factor affecting the time of fall is the vertical distance (height of the cliff) and the acceleration due to gravity. The initial horizontal velocity does not influence the time of fall. It only affects the horizontal displacement (range) of the object.

What if the Height of the Cliff is Unknown?

If the height of the cliff is unknown, you cannot calculate the time of fall. The provided equation:

S V_i t frac{1}{2} a t^2

Uses the knowledge of the height to solve for time. For example:

-H frac{1}{2} (-9.81) t^2

Where H is the height of the cliff. Solving for time, you get:

t sqrt{frac{2H}{g}}

If H is 19.6 meters, then:

t sqrt{frac{2 times 19.6}{9.81}} approx 2 , text{s}

Conclusion

The time it takes for a stone to reach the bottom of a cliff when thrown horizontally is determined by the height of the cliff and gravity. The horizontal velocity does not affect the time of fall but influences the horizontal displacement only.

For more information on related topics, you can follow our physics articles resources section. Understanding such principles is crucial for anyone pursuing a career in physics or engineering.