How to Calculate the Height an Object Will Hover: A Comprehensive Guide

When an object is subjected to a force, balancing this against the gravitational force allows us to determine how high the object will hover. This guide will walk you through the steps using the key concepts of weight, net force, and kinematic equations to calculate the hover height. We will also discuss the implications of the force's behavior with respect to height.

Key Concepts

To understand how to calculate the height an object will hover, we need to consider the following key concepts:

Weight of the Object: The weight ( W ) of an object is determined using the formula ( W m cdot g ), where ( m ) is the mass in kilograms and ( g ) is the acceleration due to gravity on Earth, which is approximately ( 9.81 , text{m/s}^2 ). Net Force: The net force is the difference between the upward force ( F ) and the gravitational force. When the upward force equals the weight, the object hovers; if it exceeds the weight, the object accelerates upwards. Accomplishing Hover: For an object to hover, the vertical force applied must balance the gravitational force acting on it. Kinematic Equations: These equations can be used to calculate the height an object will reach if the force is applied over a certain period or if the initial velocity is known.

Given Values and Calculation Steps

Let's consider an object with a mass ( m 1 , text{kg} ) and an upward force ( F 20 , text{N} ).

Calculate the Weight:

The weight of the object is calculated as follows:

[ W m cdot g 1 , text{kg} cdot 9.81 , text{m/s}^2 9.81 , text{N} ] Calculate the Net Force:

The net force is the difference between the upward force and the weight:

[ text{Net Force} F - W 20 , text{N} - 9.81 , text{N} 10.19 , text{N} ] Calculate the Acceleration:

Using Newton's second law, ( F m cdot a ), we can find the acceleration:

[ a frac{text{Net Force}}{m} frac{10.19 , text{N}}{1 , text{kg}} 10.19 , text{m/s}^2 ]

To accurately determine the height the object will hover, additional information such as the duration of the force application or initial velocity is required.

Height Calculation

If the object starts from rest and the force is applied for a certain time ( t ), the kinematic equation can be used to find the height:

[ h frac{1}{2} a t^2 ]

Where ( h ) is the height and ( a ) is the acceleration calculated earlier.

Implications and Further Considerations

For an object to hover, the applied force must balance the gravitational force. Here's a more detailed breakdown of the implications:

Hovering Condition: If the upward force exactly matches the weight, the object hovers at a constant height. Acceleration: If the upward force exceeds the weight, the object will accelerate upwards. As it rises, the gravitational force weakens, causing the object to accelerate more rapidly. Atmospheric Variation: In reality, atmospheric conditions can affect the gravitational force. For most flying machines, the force decreases with height due to the thinning atmosphere, allowing the object to hover more effectively.

In conclusion, the height an object will hover can be calculated using the given force and its weight. However, the actual height depends on the duration of the force application or initial velocity. Accounting for atmospheric conditions provides a more accurate understanding of hovering behavior in practical scenarios.