Equations of Motion
In this lesson and the next set of lessons we will look at the third way to describe motion. We have looked at describing motion in terms of words and graphs. Now, we examine equations that can be used to describe motion. These lessons are about solving ...
Equations of Motion
In this lesson and the next set of lessons we will look at the third way to describe motion. We have looked at describing motion in terms of words and graphs. Now, we examine equations that can be used to describe motion.
These lessons are about solving problems relating to uniformly accelerated motion. In other words, motion at constant acceleration.
The following are the variables that will be used in this section:
\begin{align*} \vec{v}_{i} & = \text{ initial velocity } \text{m·s$^{-1}$} \text{ at } t = \text{0}\text{ s} \\ \vec{v}_{f} & = \text{ final velocity } \text{m·s$^{-1}$} \text{ at time } t \\ \Delta \vec{x} & = \text{ displacement } \text{(m)} \\ t & = \text{ time } \text{s} \\ \Delta t & = \text{ time interval } \text{s} \\ \vec{a} & = \text{ acceleration } \text{m·s$^{-2}$} \end{align*}An alternate convention for some of the variables exists that you will likely encounter so here is a list for reference purposes:
\begin{align*} \vec{u} & = \text{ initial velocity } \text{m·s$^{-1}$} \text{ at } t = \text{0}\text{ s} \\ \vec{v} & = \text{ final velocity } \text{m·s$^{-1}$} \text{ at time } t \\ \vec{s} & = \text{ displacement } \text{(m)} \end{align*}Did You Know?
Galileo Galilei of Pisa, Italy, was the first to determined the correct mathematical law for acceleration: the total distance covered, starting from rest, is proportional to the square of the time. He also concluded that objects retain their velocity unless a force – often friction – acts upon them, refuting the accepted Aristotelian hypothesis that objects “naturally” slow down and stop unless a force acts upon them. This principle was incorporated into Newton's laws of motion (1st law).
In this book we will use the first convention.
\begin{align*} {\vec{v}}_{f} & = {\vec{v}}_{i} + \vec{a}t \qquad (1) \\ \Delta \vec{x} & = \frac{\left({\vec{v}}_{i} + {\vec{v}}_{f}\right)}{2}t \qquad (2) \\ \Delta \vec{x} & = {\vec{v}}_{i}t + \frac{1}{2}\vec{a}{t}^{2} \qquad (3) \\ {v}_{f}^{2} & = {v}_{i}^{2} + 2\vec{a}\Delta \vec{x} \qquad (4) \end{align*}The questions can vary a lot, but the following method for answering them will always work. Use this when attempting a question that involves motion with constant acceleration. You need any three known quantities (\({\vec{v}}_{i}\), \({\vec{v}}_{f}\), \(\Delta \vec{x}\), \(t\) or \(\vec{a}\)) to be able to calculate the fourth one.
Problem Solving Strategy:
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Read the question carefully to identify the quantities that are given. Write them down.
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Identify the equation to use. Write it down!!!
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Ensure that all the values are in the correct units and fill them in your equation.
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Calculate the answer and check your units.Equations of motion
This lesson is part of:
One-Dimensional Motion