How to Use the Quadratic Formula to Find the Height of a Moving Object
Understanding the motion of objects is a fundamental aspect of many fields, including physics, engineering, and even everyday life. One common scenario involves determining the height of an object at a given time when the object is in motion under the influence of gravity. This can be achieved using the quadratic formula, which is derived from the motion equations. In this article, we will explore how the quadratic formula can be utilized to find the height of a moving object.
Introduction to the Quadratic Formula
The quadratic formula is a powerful mathematical tool used to solve quadratic equations of the form:
ax2 bx c 0
where a, b, and c are constants, and x represents the variable. The formula to find the roots of such an equation is given by:
x [-b ± √(b2 - 4ac)] / 2a
This formula is widely used in various applications, but can also be applied to problems involving height and motion.
Relating Quadratic Formula to Height Calculation
To apply the quadratic formula to height calculations, it is necessary to first understand the motion equations. Consider an object thrown vertically upwards or falling downwards under the influence of gravity. The vertical position of the object as a function of time can be described by the following equation:
y -1/2 gt2 vt y0
where:
y is the height of the object at time t, g is the acceleration due to gravity (approximately 9.81 m/s2 on Earth), v is the initial velocity of the object, y0 is the initial height of the object.This equation is a quadratic in t (time). It can be rearranged into the standard form of a quadratic equation to facilitate the use of the quadratic formula.
Step-by-Step Guide to Finding the Height Using the Quadratic Formula
Step 1: Establish the Quadratic Equation
Given the motion equation y -1/2 gt2 vt y0, it can be rewritten in the standard quadratic form as:
at2 bt c 0, where a -1/2 g, b v, and c y0.
Step 2: Apply the Quadratic Formula
Substitute the identified values of a, b, and c into the quadratic formula:
t [-b ± √(b2 - 4ac)] / 2a
This will provide the times at which the object reaches the desired height.
Example: Calculating the Height at a Specific Time
Let's take an example. Suppose an object is thrown vertically upwards with an initial velocity of 20 m/s from a height of 50 meters. We want to find the height at which the object will be at 3 seconds into its journey.
Step 1: Identify the Constants
- g 9.81 m/s2, the acceleration due to gravity - v 20 m/s, the initial velocity - y0 50 m, the initial height
Hence, the quadratic equation will be:
-4.905t2 20t 50 0
Here, a -4.905, b 20, and c 50.
Step 2: Calculate the Time
Using the quadratic formula:
t [-20 ± √(202 - 4(-4.905)(50))] / 2(-4.905)
t [-20 ± √(400 981)] / -9.81
t [-20 ± √1381] / -9.81
t [-20 ± 37.17] / -9.81
This gives two solutions:
t (17.17 / -9.81) ≈ -1.75 ± 3.83
The positive value is the relevant solution, as time cannot be negative. Therefore, the height at 3 seconds is:
y -4.905(3)2 20(3) 50 ≈ 64.58 m
Conclusion
The quadratic formula is a versatile tool for solving a variety of problems, including determining the height of a moving object. By understanding the motion equations and applying the quadratic formula, one can accurately calculate the height at any given time for objects thrown vertically or falling under the influence of gravity. This method is widely used in physics, engineering, and real-world scenarios, making it an essential skill for professionals and enthusiasts alike.