### Unit 2: Kinematics in a Straight Line

We begin our formal study of physics with an examination of kinematics, the branch of mechanics that studies motion. The word "kinematics" comes from a Greek term that means "motion". Note that kinematics is not concerned with what causes the object to move or to change course. We will look at these considerations later in the course. In this unit, we examine the simplest type of motion, which is motion along a straight line or in one dimension.

**Completing this unit should take you approximately 5 hours.**

Upon successful completion of this unit, you will be able to:

- compare and contrast distance and displacement;
- define and distinguish between vector and scalar physical quantities;
- explain the relationship between instantaneous and average values for physical quantities;
- compare and contrast speed and velocity;
- solve one-dimensional kinematics problems;
- describe the effects of gravity on an object in motion;
- calculate the position and velocity of an object in free fall; and
- draw and interpret graphs for displacement and velocity as functions of time, and determine velocity and acceleration from them.

### 2.1: Vectors, Scalars, and Coordinate Systems

A scalar physical quantity is a measurement of quantity that has a magnitude (amount), but not a direction. Examples of scalar quantities include mass and temperature; no direction is associated with these measurements. Distance is also a scalar quantity because it has no direction associated with it.

A vector physical quantity is a measurement that has a magnitude (amount) and direction. Vectors are often depicted as an arrow. The length of the arrow shows the magnitude of the quantity, and the direction of the arrow shows the direction of the vector.

For simple one-dimensional systems, a vector is often written as the magnitude with a (+) or (−) to indicate direction, with (+) going toward the right and (−) going toward the left. Displacement and velocity are examples of vector quantities. For example, 5.5 km/s east. This measurement shows the magnitude of the velocity (5.5 km/s), and the direction (east).

Read this text to learn more about the difference between a scalar and vector quantity.

### 2.2: Instantaneous and Average Values for Physical Quantities

An instantaneous value is a value measured at a given instant, or time. For example, we can measure the velocity of an object right at high noon as 5.5 km/s east. This is an instantaneous value because we measured it at a given instant in time. A car's speedometer is an example of an instantaneous measurement. Furthermore, velocity does not necessarily stay constant over time, so instantaneous measurements can vary depending on when you take the measurement.

An average value is calculated over a period of time. For example, to calculate average speed, divide the distance traveled by time traveled. For example, if you drive 30 miles in two hours, your average speed is 15 miles/hour. However, as we know from driving, we rarely drive exactly the same speed for two hours. So, the instantaneous value of your speed could vary at any given time, but the average value is still 15 miles/hour.

Read this text which explains what we mean when we talk about instantaneous versus average time, velocity, and speed. Note that we will explore the calculations you see in this text in detail in Section 2.4 below. For now, make sure you understand what these concepts mean before we begin discussing distance and displacement next.

### 2.3: Distance and Displacement

Distance describes how much an object has moved. It depends on how the object has moved, that is, the path the object took to get from the starting point to the ending point. The units for distance pertain to length, such as meters. Distance is a scalar quantity because it describes the magnitude of the measurement, but not a specific direction.

Displacement describes an object's overall change in position. It only depends on the starting and ending points of the object. It does not depend on the path taken to get between the two points. Like distance, the units for displacement are also length, such as meters. However, displacement is a vector quantity, which means it has a magnitude and a specific direction associated with the measurement. So, the complete value for displacement must also include a direction.

For an example, consider a four-story building. A person needs to travel on the elevator from the first to the third floor. To accomplish this, the person could take an elevator directly from the first floor to the third floor. In this case, the distance and displacement are the same, because the person went directly from the starting to the ending point.

However, this is not the only way the person could travel from the first to the third floor. They could accidentally hit the fourth floor button when they got on the elevator. In this case, they would travel from the first floor to the fourth floor, and back down to the third floor. In this instance, the displacement is still from the first floor to the third floor. But, the distance is longer, because the person took a detour to the fourth floor before going back down to the third floor.

Read this text, which discusses position and the difference between distance and displacement.

This lecture accompanies what you just read.

This video demonstrates the difference between distance and displacement, and their associated graphs with time.

Watch this video to learn how to differentiate distance and displacement, and graph them with respect to time.

Watch this video to review scalars, vectors, and how they connect with distance and displacement.

### 2.4: Speed and Velocity

Elapsed time, , is the change in time. Elapsed time is calculated as , where is final time and is initial time. The Greek letter delta, Δ, means change. So, Δt means change in time. You will see this frequently in this course. When calculating elapsed time, we often assume the initial time is zero, to make the subtraction easier.

Average velocity is the displacement divided by the elapsed time: . Here, the line above the shows that it is an average quantity. This is the common notation for average quantities. To calculate the average velocity, divide the change in displacement by the elapsed time.

The average velocity is a vector quantity because displacement is a vector quantity. Because we calculate average velocity from a vector quantity, it itself is a vector quantity. This means that average velocity has a direction associated with it. In one-dimensional systems, this means that the average velocity is written with a (+) or (−) sign, depending on the direction of the displacement.

Instantaneous speed is the magnitude of the instantaneous velocity, measured at a given time or instant. Unlike velocity, instantaneous speed is a scalar quantity, so it does not have a direction associated with it. For example, if the instantaneous velocity of an object is −2 m/s in one-dimensional motion, the object's instantaneous speed is simply 2 m/s.

The average speed of an object is the object's distance divided by the elapsed time. This is similar to the average velocity, which is the object's displacement divided by the elapsed time. Recall that distance is a scalar quantity that describes how much an object moved and that it can be very different from the vector displacement. Therefore, the average speed of an object is also a scalar quantity, and it can differ from the average velocity.

Now, let's return to this text which you read in Section 2.2, to study the calculations in more detail. Pay attention to Figure 2.10 and Figure 2.11 which show examples of how to calculate displacement, distance, average speed, and velocity.

This lecture accompanies what you just read.

Watch this video to discover how average speed and velocity relate to distance and displacement.

### 2.5: Motion with Constant Acceleration

Acceleration (𝑎) is the rate of change of velocity. We can calculate the average acceleration using the following equation:

Because velocity is a vector, acceleration is also a vector quantity. Instantaneous acceleration is acceleration measured at a specific instant in time. In most kinematic problems, we assume average acceleration is a constant value.

Read this text. Pay attention to the examples which show how to solve equations of motion. These include how to calculate displacement, given average velocity and time, and how to calculate final velocity, given initial velocity, acceleration, and time.

This lecture accompanies what you just read.

Read this text for more examples and practice on how to solve motion equations for constant acceleration.

- Watch this video to get a better understanding of acceleration and how it is graphed with respect to time.
- Watch this video to learn how to analyze acceleration with equations.

### 2.6: Falling Objects

Gravity is a force that attracts objects toward the center of the earth, or more generally speaking, massive objects to one another. In the absence of friction or air resistance, all objects fall with the same acceleration toward the center of the earth. This is known as free-fall. The acceleration due to gravity is .

In reality, air resistance affects the acceleration of falling objects. Air resistance opposes the motion of an object in air, and causes falling lighter objects to accelerate less than heavier objects. This is why a feather falls to earth slower than a heavier object like a brick. If there was no air resistance, a feather and brick would fall to earth with the same acceleration due to gravity.

Read this text for more about falling objects, gravity, and freefall. We will discuss how to calculate the examples here in the next section.

This lecture accompanies what you just read.

Watch this live demonstration which takes place at zero gravity at the end of the last Apollo 15 moonwalk.

### 2.7: Calculating the Kinematic Quantities of Objects in Constant Acceleration

To perform calculations involving objects in constant acceleration situations, such as free fall, we first need to use the basic definitions of velocity and acceleration to derive useful formulas called "kinematic equations".

We can use kinematic equations for any situation where there is a constant acceleration acting on an object (including zero acceleration), and included with this situation is freefall. In free fall, acceleration (a) equals the acceleration due to gravity (g). For an object falling, we use −g to show the vector's downward direction of free fall.

As you read, pay attention to the relevant equations in the box Kinematics Equations for Objects in Free Fall where Acceleration = −g

Note that because the motion is free fall, a is simply replaced with (here, is the acceleration due to gravity, ) and the direction of motion is the direction, rather than the direction. When calculating the position and velocity of an object in freefall, we need to consider two different conditions. First, the object can be thrown up as it enters freefall. For example, you could throw a baseball up and watch it fall back down.

Complete the steps in Example 2.14. After you review the solution, pay attention to the graphs in Figure 2.40. You can throw an object directly downward as it enters freefall, such as when you throw a baseball directly down from a second-floor window.

Then, complete the steps in Example 2.15. Notice that Figure 2.42 compares what is happening in Example 2.14 and Example 2.15. It is important to understand the difference between an object that is thrown up and enters free fall, versus an object that is directly thrown down. We can often use experimental data to calculate constants, such as .

In Example 2.16, we determine the acceleration due to gravity constant () from experimental data.

Watch this video for details on how kinematic equations were derived.

Watch this video for details on kinematic equations and its relationship with freefall.

Read this text for a step-by-step guide on how to solve problems of constant acceleration using kinematic equations.

This lecture accompanies what you just read.

- This video goes into more detail about constant acceleration.

### 2.8: Graphical Analysis

When graphing two variables against each other, we generally define the dependent variable as the variable on the vertical axis (y-axis) and the independent variable as the variable on the horizontal axis (x-axis). When plotting a straight line, we use the equation , where is the slope and is the y−intercept of the line.

We define slope as:

The y−intercept is the point where the line crosses the y-axis of the graph.

Read this text for an introduction on graphical position, velocity, and acceleration with regards to one another. As you read, pay attention to Figure 2.46 which is an example of a linear graph is the graph of position versus time when acceleration is zero.

See an example of this type of graph in Figure 2.47. In this graph, we can determine the slope by picking two different points on the line, taking the change in y-value, and dividing it by the change in x-value between those two points. In this case, the unit for slope is m/s, which is the unit for velocity. Therefore, the slope for a graph of position versus time with zero acceleration is the average velocity of that object. See how to calculate the average velocity of an object from this type of graph in Example 2.17.

When acceleration is a non-zero constant, the graph of position versus time is no longer linear. You can see an example of this type of graph in Figure 2.48. Note that while the position versus time graph is not linear, the velocity versus time graph is linear. In the position versus time graph, the slope at any given point is the instantaneous velocity of the object. The instantaneous slope can be determined by drawing a tangent line at the desired point along the graph and determining slope. Pay attention to the tangent lines drawn in Figure 2.48 (a).

To determine instantaneous velocity at a given time when acceleration is a non-zero constant, take a look at Example 2.18. We can determine instantaneous velocities at multiple points along a position-time graph with constant non-zero acceleration and make a table relating these instantaneous velocities to the specified time along the x-axis where we found them. Then, we can use that table to plot velocity versus time. This process is demonstrated in Figure 2.48 (a) and (b). The slope of this linear graph has units , which are acceleration units. Therefore, the slope of the velocity versus time graph is acceleration.

- Watch this video for details regarding the velocity versus time graphs and how they relate to acceleration.

### Unit 2 Assessment

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