Tru Physics

Chapter 4: Position, Velocity, and Acceleration

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4.1 Position and Displacement

Position is a fundamental concept in physics that describes an object’s location in space relative to a reference point. To understand position, we need to define a few key terms. The first is distance, which is the total length of the path an object travels. This can be a bit tricky, as the path an object takes may not be a straight line. For example, if you go for a walk around your neighborhood, the distance you cover will be greater than the straight-line drawn between your starting point and ending point.

The second key term is displacement, which is the change in an object’s position. It is calculated by subtracting the initial position from the final position. Displacement is a vector quantity, meaning it has both magnitude (how far an object moved) and direction (the direction in which it moved). For example, if you walk from your starting point to the end of the block and then turn around and walk back to your starting point, your displacement will be zero because your final position is the same as your initial position. Displacement is written as \Delta x and is given by the following formula:

\Delta \vec{x} = \vec{x}_f - \vec{x}_i

In physics, we often use coordinate systems to describe an object’s position. The most common coordinate system is the Cartesian coordinate system, which uses two or three axes to describe position in two or three dimensions, respectively. For example, if we use the Cartesian coordinate system to describe the position of a point in two dimensions, we would give its x-coordinate (its position on the horizontal axis) and its y-coordinate (its position on the vertical axis).

4.2 Speed and Velocity

Speed and velocity are two concepts that are often used interchangeably in everyday life but are actually very different. Speed is a scalar quantity that describes how fast an object is moving, whereas velocity is a vector quantity that describes both the speed and direction of an object’s motion. Both have units of meters per second (\text{m}/\text{s}). Put simply, velocity is speed with a specified direction. Velocity can be calculated using the formula below:

\vec{v}_\text{avg} = \dfrac{\Delta \vec{x} }{\Delta t}=\dfrac{\vec{x}_f-\vec{x}_i}{t_f-t_i}

For example, imagine you are driving in your car on a straight road. If you are driving 60 miles per hour, you have a speed of 60 mph. However, if you are driving 60 mph east, your velocity is 60 mph to the east.

It’s also important to discuss the relation between velocity and position. Velocity is the slope of the position vs. time (x vs. t) graph. It can be an average velocity if the slope is take over some time interval. It can also be an instantaneous velocity if the slope is taken over an infinitesimally small interval such that \Delta t \to 0. The instantaneous velocity in some x-direction can be expressed as:

v_x = \displaystyle\lim_{\Delta t \to 0}\dfrac{\Delta x}{\Delta t}=\dfrac{dx}{dt}

where d/dt is the derivative with respect to time. Taking derivatives is a calculus concept that should be studied before moving on to the next chapter. Power rule derivatives are generally sufficient for most physics 1 applications.

position, velocity, acceleration. Speed and velocity are concepts familiar in everyday life.
Speed and velocity are concepts familiar in everyday life.

4.3 Acceleration

Acceleration is a change in velocity, or the rate at which an object’s velocity changes over time. Acceleration can be positive, negative, or zero, depending on whether the object is speeding up, slowing down, or moving at a constant velocity. To calculate acceleration, we use a formula of the same form as the one used above to calculate velocity.

\vec{a}_\text{avg} = \dfrac{\Delta \vec{v}}{\Delta t}=\dfrac{\vec{v}_f-\vec{v}_i}{t_f-t_i}

Intuitively, acceleration can be thought of as a change in speed. For example, imagine you are sitting in a car that is stopped at a red light. Consider forward to be the positive direction and backward to be the negative direction. When the light turns green, the car starts to move forward. This is a positive acceleration because the speed of the car is increasing and the direction of motion is forward. Conversely, if you are driving in your car and begin applying the brakes, you experience negative acceleration because your speed (in the positive direction) is decreasing.

Acceleration is a vector quantity, like velocity, and it has units of meters per second squared (\text{m}/\text{s}^2). This means that acceleration is more accurately defined as a measure of how quickly the velocity of an object changes over time.

Just like velocity, we can define aceleration to be both an average value and an instantaneous value. The instantaneous value also allows the limit of \Delta t \to 0. Thus, we can write the instantaneous acceleration in some x-direction as:

a_x=\displaystyle\lim_{\Delta t \to 0}\dfrac{\Delta v}{\Delta t}=\dfrac{dv}{dt}

with d/dt the derivative with respect to time.

4.4 Position, Velocity, and Acceleration Equations

Let’s summarize everything we have so far in terms of equations. Let’s suppose that all of the motion is taking place in a straight line—call it the x-direction. It doesn’t really matter what direction it is because you as the physicist get to determine how you position your coordinate system.

4.4.1 Position and Displacement

Let’s start with position. Position is just a point, so you can’t really make an equation out of it. However, what if you have a change in position (\Delta x)? Well, a change in position is called a displacement. Thus, the displlacement of an object is given mathematically as the difference between the final and intial positions:

\Delta \vec{x} = \vec{x}_f - \vec{x}_i

We know that the inverse operation of a derivative is an integral. Thus, by reversing the equation in 4.4.2 below, we can retrieve another equation for \Delta x.

\Delta x = \displaystyle\int_{t_i}^{t_f}v(t)dt

4.4.2 Average and Instantaneous Velocity

Velocity is fundamentally different from position because it is a rate of change. The word rate implies that there is a change in time involved here. Specifically, we want to know what change in position happens within a certain change in time. Without calculus, we can calculate an average velocity using the slope of a position vs time (x vs. t) graph over some time interval. Mathematically, this corresponds to:

\vec{v}_\text{avg} = \dfrac{\Delta \vec{x} }{\Delta t}=\dfrac{\vec{x}_f-\vec{x}_i}{t_f-t_i}

On the other hand, with calculus, we can shrink the interval of \Delta t until is infinitely small. This leads us to the derivative with respect to time, and we can express the instantaneous velocity then as:

v_x = \displaystyle\lim_{\Delta t \to 0}\dfrac{\Delta x}{\Delta t}=\dfrac{dx}{dt}

Once again, knowing that the inverse of the derivative operation is an integral, we can look at the equation in 4.4.3 below and retrieve another equation for \Delta v.

\Delta v = \displaystyle\int_{t_i}^{t_f}a(t)dt

4.4.3 Average and Instantaneous Acceleration

Acceleration is a rate of change, just like velocity. However, instead of giving the rate of change of position, it gives the rate of change of velocity. Acceleration is to velocity what velocity is to position. These three are all intimately related. Once again, without calculus, we can calculate an average acceleration as the slope of the velocity vs time graph over some time interval. Mathematically, this corresponds to:

\vec{a}_\text{avg} = \dfrac{\Delta \vec{v}}{\Delta t}=\dfrac{\vec{v}_f-\vec{v}_i}{t_f-t_i}

Just as before, we can shrink the interval of \Delta t until is infinitely small. This leads us to the derivative with respect to time, and we can express the instantaneous acceleration then as:

a_x=\displaystyle\lim_{\Delta t \to 0}\dfrac{\Delta v}{\Delta t}=\dfrac{dv}{dt}

Theoretically, you could derive another equation to calculate \Delta a in the same way that was done previously for \Delta x and \Delta v. This is generally not needed for most physics 1 applications, though disciplines involving robotics and engineering may find it useful. We will not use such a definition in this course.

Chapter Summary

Displacement is a vector quantity representing a change in position. Speed and velocity are two different concepts that describe how fast an object is moving. Speed is a scalar quantity that only describes magnitude, whereas velocity is a vector quantity that describes both the speed and direction of an object’s motion. Acceleration is a change in velocity, or the rate at which an object’s velocity changes over time, and it has units of meters per second squared (\text{m}/\text{s}^2). By understanding these concepts, we can better understand the motion of objects in our everyday experiences.

position, velocity, acceleration. A motorcyclist accelerates around a turn.
A motorcyclist accelerates around a turn.

Enroll on Canvas

This course uses Canvas for homework assignments, quizzes, and exams. These assignments are open to everyone. Anyone is allowed to enroll in the Canvas course. In fact, this is highly encouraged as it will help you track your progress as you go through the course. Graded feedback will help you get an idea for what your grade would actually be in a Physics 1 college course. Use this link to enroll in the Canvas course.

Quiz 2 is also available and covers content from Chapters 3 and 4. Enroll in the Canvas course using the link above if you haven’t already. Then, click here to take the quiz.

Knowledge Check

Answer the quiz questions below.

What is the difference between distance and displacement?
Distance is a vector quantity, while displacement is a scalar quantity.
No, it’s the other way around. Distance is a scalar quantity which is the total length of the path an object travels, while displacement is a vector quantity that describes the change in an object’s position and has both magnitude and direction.
Distance and displacement are the same thing.
Incorrect, distance is the total length of the path an object travels, while displacement is the change in an object’s position, taking into account both magnitude and direction.
Distance is the total length of the path an object travels, while displacement is the change in an object’s position and has both magnitude and direction.
Correct! Distance describes the total path an object travels without taking into account the direction, while displacement is a vector quantity that shows how far an object moved and in which direction.
What is the difference between speed and velocity?
Speed is a vector quantity, while velocity is a scalar quantity.
No, it’s the other way around. Speed is a scalar quantity that describes how fast an object is moving, while velocity is a vector quantity that describes both the speed and direction of an object’s motion.
Speed and velocity are the same thing.
Incorrect, speed is a scalar quantity describing how fast an object is moving, while velocity is a vector quantity that includes both speed and direction of an object’s motion.
Speed is a scalar quantity that describes how fast an object is moving, whereas velocity is a vector quantity that describes both the speed and direction of an object’s motion.
Correct! Speed only tells us how fast an object is moving, whereas velocity tells us how fast and in which direction an object is moving.
How can acceleration be defined?
It is the rate of change of position.
Incorrect, acceleration is the rate of change of velocity, not position.
It is the distance travelled per unit of time.
No, that definition refers to speed, not acceleration. Acceleration is the rate of change of velocity.
It is the rate at which an object’s velocity changes over time.
Exactly! Acceleration is a measure of how quickly the velocity of an object changes over time. It can be positive, negative, or zero, depending on whether the object is speeding up, slowing down, or moving at a constant velocity.
If the displacement of an object is zero, what can we conclude about its motion?
The object has moved a large distance.
No, a displacement of zero actually indicates that the object’s initial and final positions are the same, regardless of the distance traveled.
The object’s speed is zero.
Not necessarily. Even if the displacement is zero, the object might have moved at a considerable speed before returning to its starting position.
The object’s initial and final positions are the same.
Exactly! A displacement of zero means that the object has returned to its initial position, regardless of the path taken or distance covered.
Continue to Chapter 5: Constant Velocity
Back to Chapter 3: Vector Components

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4 responses to “Chapter 4: Position, Velocity, and Acceleration”

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