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 and is given by the following formula:
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 Put simply, velocity is speed with a specified direction. Velocity can be calculated using the formula below:
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 The instantaneous velocity in some x-direction can be expressed as:
where 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.
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.
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 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 Thus, we can write the instantaneous acceleration in some x-direction as:
with 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 ? 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:
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
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:
On the other hand, with calculus, we can shrink the interval of until is infinitely small. This leads us to the derivative with respect to time, and we can express the instantaneous velocity then as:
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
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:
Just as before, we can shrink the interval of until is infinitely small. This leads us to the derivative with respect to time, and we can express the instantaneous acceleration then as:
Theoretically, you could derive another equation to calculate in the same way that was done previously for and 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 . By understanding these concepts, we can better understand the motion of objects in our everyday experiences.
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