Visual models of motion

What is an object? The answer might seem obvious. However, when you actually try to define an “object” you may find that it’s not so straightforward. For instance, a water bottle: object, right? But, what if the water bottle has a cap? Is that a separate object? What if it has ice cubes in it? Are they part of the water bottle object or are they separate objects?

For the purposes of representing motion, it is useful to define an object as something for which the internal structure is not important to the situation. When the quantities of interest are not affected by the object’s size, shape, or internal structure, we can choose to ignore them. This simplifies the situation because the object can be represented as a single dot. When we do this, we are using the object model.

[[☃ explanation 1]]

##Using the object model

Suppose a car is waiting at a red light. The light turns green and the car begins to speed up to the right. We want to represent this motion visually and simply. One way to do this is to draw a particle diagram. The particle is the dot that represents the car. In this case, we can treat the car as an object because we just care about representing its overall motion, not the motion of anything on or inside of it. We shrink the car to a single point.

[[☃ image 1]]

Now we can use the dot to show the position of the car at regular intervals of time. This is called a particle diagram. (You may also see this called an “oil drop diagram.”)

[[☃ image 6]]

From the diagram above, we can infer that the car is speeding up. The dots show where the car is after equal amounts of time. Since the car travels more and more distance in each successive time interval, it must be speeding up.

The particle diagram we just made is a qualitative diagram and is useful for visualizing how the position of the car is changing over time. But, it doesn’t give us any quantitative (numeric) information. For that, we need to add a reference frame and define the elapsed time between each successive set of dots, as shown below:

[[☃ image 12]]

Now we have quantitative information! We can see that the first dot is at a position of $0,\text{m}.$ Since the starting time is not specified in this case, we get to decide its value. Let’s call the first time $0,\text{s}.$ The diagram is labeled with the elapsed time between dots, $\Delta{t}=2,\text{s}.$ So, using the number line, we can determine the car's approximate position, $x,$ every $2,\text{s}.$ This information can be summarized in a data table:

We could go on this way determining the position of the car every $2,\text{s}$ if more dots were shown. It can also be helpful to label the time at which each position is shown on the diagram:

[[☃ image 10]]

##Motion diagrams

Motion diagrams combine particle diagrams with vector arrows to represent various motion quantities. A common quantity to show on a motion map is velocity. Velocity is the change in position per unit of time. So, the spacing between the dots can be used to help draw the velocity vectors to scale. Our particle diagram of the car is shown below with velocity vectors added, making it a motion diagram:

[[☃ image 4]]
Notice that “$v$” appears above each arrow to denote velocity. This is necessary because motion maps can also show other vector quantities, like acceleration. Acceleration and velocity are different quantities so they will be represented by different vectors.

##Try it!

###Example 1

Describe the motion of the object represented on the motion diagram below.

[[☃ image 1]]

[[☃ explanation 1]]

###Example 2

Describe the motion of the object represented on the motion diagram below.

[[☃ image 2]]

[[☃ explanation 2]]

##Common misconceptions

• The “first” dot on a motion diagram represents the position of the object at the earliest time shown. This is not necessarily the one on the left.

• Each dot on a motion map represents the position of an object at a particular instant of time. In contrast, the time interval is the period of time between each successive set of dots. Typically the first dot represents the position at time $t=0.$ Therefore, if the intervals between dots are $1,\text{s}$ long, the second dot will be at time $t=1,\text{s}$, the third dot at time $t=2,\text{s}$, and so on.

A position vs. time graph represents the position of an object as a function of time.

For example, if you read the $y$-value of the graph below at a particular time (in seconds), you will get the position of the object (in meters) at that instant.

Try sliding the dot horizontally on the graph below to see the position at different times.

[[☃ interaction 1]]

Concept check: According to the graph above, what is the object's position at time $t=5\text{ s}?$

[[☃ explanation 1]]

##What does the slope of a position vs. time graph represent?

The slope of a position vs. time graph represents the velocity of the object.

To see why, consider the graph below, which represents the position, $x,$ of an object moving in one dimension as a function of time, $t$:

[[☃ image 1]]

The slope of the graph is:

$\text{slope}=\dfrac{\text{rise}}{\text{run}}=\dfrac{\vec{x}_2-\vec{x}1}{t_2-t_1}=\dfrac{\Delta \vec{x}}{\Delta t}=\vec{v}{avg}$

The slope of a position vs. time graph represents the change in the object's position divided by the time taken for that change. That's average velocity!

The graph above is a straight line—it has constant slope, so we know the object moved with constant velocity. Therefore the average velocity is the same as the velocity at any given time. But what if an object does NOT move with constant velocity? In that case, we should expect the slope of its position vs. time graph to change.

Consider the example below. We can tell right away that this object's velocity changed (it accelerated) over the time interval shown, since the slope changes. To know what the velocity was at a given instant—the instantaneous velocity—we need the slope at that instant. In other words, we need the slope of the line tangent to the curve at that time.

Try sliding the dot horizontally on the graph below to see the slope of the tangent line at different times.

[[☃ interaction 2]]

Concept check: At what time(s) did the object have an instantaneous velocity of $v=0\text{?}$

[[☃ explanation 4]]

Concept check: From $t=0\text{ s}$ to $t=3 \text{ s}$ was the object moving in the $+$ or $-$ direction? Did it speed up, slow down, or both?

[[☃ explanation 5]]

Concept check: From $t=3\text{ s}$ to $t=9 \text{ s}$ was the object moving in the $+$ or $-$ direction? Did it speed up, slow down, or both?

[[☃ explanation 6]]

##Average velocity

Above, we considered the instantaneous velocity of an object, equal to the slope of its position vs. time graph at an instant. But sometimes, we want to know the average velocity of an object over an interval.

On a position vs. time graph, the average velocity for an interval is equal to the slope of the line connecting the endpoints of the interval.

Consider the position vs. time graph below.

[[☃ image 2]]

Concept check: What was the average velocity of the object from $t=0\text{ s}$ to $t=9\text{ s}\text{?}$

[[☃ explanation 8]]

Concept check: What was the average velocity of the object from $t=4\text{ s}$ to $t=9\text{ s}\text{?}$

[[☃ explanation 9]]

Concept check: What was the average velocity of the object from $t=6\text{ s}$ to $t=9\text{ s}\text{?}$

[[☃ explanation 10]]

##Common mistakes and misconceptions

• The position vs. time graphs above represent motion in one dimension. The object did NOT trace out the shape on the graph. For example, if a graph goes up and then down, that doesn't necessarily mean the object went up and then down. The object could have gone forward and then backward. Do not think of the shape of the graph as the object's path. Instead, it is a record of the object's position as a function of time.

• If the y-value of the graph is positive (the line is above the x-axis), the object was located at a positive position at that time. If the slope of the graph is positive (tilted up), the object had positive velocity at that time. At any time the position and velocity may both be positive, both be negative, or have different signs. It's very easy to confuse them. Remember to only consider slope when determining velocity.

• The average velocity over an interval may or may not equal the instantaneous velocity at all times throughout that interval.

A velocity vs. time graph represents the velocity of an object as a function of time.

For example, if you read the $y$-value of the graph below at a particular time (in seconds), you will get the velocity of the object (in meters per second) at that instant.

**Try sliding the dot horizontally on the graph below to see the velocity at different times.**

[[☃ interaction 1]]

**Concept check:** According to the graph above, what is the object's velocity at time $t=4\text{ seconds}\text{?}$

[[☃ explanation 1]]

##What does the slope of a velocity vs. time graph represent?

The slope of a velocity vs. time graph represents the *acceleration* of the object. To see why, consider the slope of the velocity vs. time graph below:

[[☃ image 1]]

$\text{slope}=\dfrac{\text{rise}}{\text{run}}=\dfrac{\vec{v}_2-\vec{v}_1}{t_2-t_1}=\dfrac{\Delta \vec{v}}{\Delta t}=\vec{a}_{avg}$

The slope of a velocity vs. time graph represents the change in the object's velocity divided by the time taken for that change. That's average acceleration!

The graph above is a straight line—it has *constant slope*, so we know the object moved with *constant acceleration*. But what if an object does NOT move with constant acceleration? In that case, we should expect the slope of its velocity vs. time to *change*.

Consider the example below. We can tell right away that this object's acceleration changed over the time interval shown, since the slope changes. To know what the acceleration was at a given *instant*—the **instantaneous acceleration**—we need the slope at that instant. In other words, we need the slope of the line **tangent** to the curve at that time.

**Try sliding the dot horizontally on the graph below to see the slope of the tangent line at different times.**

[[☃ interaction 1]]

**Concept check:** At what time(s) did the object have a velocity of $v=0\text{?}$

[[☃ explanation 2]]

**Concept check:** At what time(s) did the object have an acceleration of $a=0\text{?}$

[[☃ explanation 4]]

**Concept check:** During what time interval was the object moving in the $+$ direction?

[[☃ explanation 3]]

**Concept check:** Is the object speeding up or slowing down at time $t=4\text{ s}$?

[[☃ explanation 1]]

##Average acceleration

Above, we considered the instantaneous acceleration of an object, equal to the slope of its velocity vs. time graph at an *instant*. But sometimes, we want to know the **average acceleration** of an object over an *interval*.

On a velocity vs. time graph, the average acceleration for an interval is equal to the slope of the line *connecting the endpoints of the interval*.

**Concept check:** On the velocity vs. time graph above, what was the average acceleration of the object from $t=0\text{ s}$ to $t=6\text{ s}\text{?}$

[[☃ explanation 1]]

##What does the area under a velocity vs. time curve represent?

The area under a velocity vs. time curve represents the *displacement* of the object. To see why, consider the velocity vs. time graph below:

[[☃ image 1]]

The graph shows an object maintaining a constant velocity of $6\text{ m/s}$ for $5\text{ s}.$

If we multiply both sides of the average velocity equation, $\vec{v}_{avg}=\dfrac{\Delta \vec{x}}{\Delta t}$, by the change in time, $\Delta t$, we get $\Delta \vec{x}=\vec{v}_{avg} \Delta t$.

By plugging in the velocity $6\text{ m/s}$ and the time interval $5\space\text{s},$ we can find the change in position (displacement):

$\Delta \vec{x}=\vec{v}_{avg}\Delta t=(6\text{ m/s})(5\text{ s})=30\text{ m}$

Now, let's see why this is equivalent to finding the area under the curve. Consider the shaded area below the curve:

[[☃ image 2]]

The area of this rectangle can be found by multiplying height of the rectangle, $6\text{ m/s,}$ by its width, $5\text{ s}:$

$\text{ area}=\text{height}\cdot\text{width} = (6\text{ m/s})(5\text{ s})=30\text{ m}$

This is the same answer we got above for the displacement.

The area under a velocity vs. time curve, *regardless of the shape*, will equal the object's displacement during that time interval.

[[☃ explanation 1]]

[[☃ explanation 2]]

##Practice

###Example 1

A windsurfer travels along a straight line. Her motion is represented on the velocity vs. time graph below.

[[☃ image 4]]

[[☃ graded-group 1]]

[[☃ explanation 1]]

###Example 2

A go-kart moves along a straight line. The go-kart's velocity as a function of time is shown below.

[[☃ image 5]]

**What was the acceleration of the go-kart at time $t=4\text{ s}$?**

[[☃ explanation 2]]

**What was the displacement of the go-kart from $t=0\text{ s}$ to $t=7\text{ s}$?**

[[☃ explanation 3]]

##Common mistakes and misconceptions

In many cases, position vs. time and velocity vs. time graphs need to be analyzed simultaneously. So, it's important to pay attention to which type of graph you're analyzing and the different meanings of each.

On a position vs. time graph, the $y$-value represents *position* and the slope represents *velocity*.

On a velocity vs. time graph, the $y$-value represents *velocity* and the slope represents *acceleration*. The area under the curve represents *displacement*.

Keeping careful track of units, both on the $y$-axis and in calculations, can help you remember what quantity you're dealing with.

An acceleration vs. time graph represents the acceleration of an object as a function of time.

For example, if you read the $y$-value of the graph below at a particular time (in seconds), you will get the acceleration of the object (in meters per second squared) at that instant.

**Try sliding the dot horizontally on the graph below to see the acceleration at different times.**

[[☃ interaction 1]]

**Concept check:** According to the graph above, what is the object's acceleration at time $t=4\text{ s?}$

[[☃ explanation 1]]

##What does the area under an acceleration vs. time graph represent?

The area under the acceleration vs. time curve for a certain time interval represents the object's *change in velocity* during that time interval. To see why, consider the acceleration vs. time graph below:

[[☃ image 3]]

The graph shows an object maintaining a constant acceleration of $4\text{ m/s}^2$ for a time of $9\space\text{s}.$

If we multiply both sides of the average acceleration equation, $\vec{a}_{avg}=\dfrac{\Delta \vec{v}}{\Delta t}$, by the change in time, $\Delta t$, we get $\Delta \vec{v}=\vec{a}_{avg} \Delta t$.

By plugging in the acceleration $4\text{ m/s}^2$ and the time interval $9\space\text{s},$ we can find the change in velocity:

$\Delta \vec{v}=\vec{a}_{avg}\Delta t= (4\text{ m/s}^2)(9\text{ s})=36\text{ m/s}$

Now, let's see why this is equivalent to finding the area under the curve. Consider the shaded area below the curve:

[[☃ image 1]]

The area of this rectangle can be found by multiplying height of the rectangle, $4\text{ m/s}^2,$ by its width, $9\text{ s}:$

$\text{area}=4\text{ m/s}^2\times 9\text{ s}=36\text{ m/s}$

This is the same answer we got above for the change in velocity.

The area under an acceleration vs. time curve, *regardless of the shape*, will equal the object's change in velocity during that time interval.

##Practice

###Example 1

An object is moving with an initial velocity of $20\text{ m/s}$ at time $t=0\text{ s}.$ The object accelerates. The acceleration of the object as a function of time is shown on the graph below.

[[☃ image 4]]

**What is the object's velocity at $t=8\text{ s?}$**

[[☃ explanation 1]]

###Example 2

A sailboat is moving with an initial velocity of $10\text{ m/s}$ at time $t=0\text{ s}.$ The blowing wind causes the sailboat to accelerate over time, as shown on the graph below.

[[☃ image 2]]

**What is the sailboat's velocity at $t=9\text{ s?}$**

[[☃ explanation 2]]

##Common mistakes and misconceptions

In many cases, position vs. time, velocity vs. time, and acceleration vs. time graphs need to be analyzed simultaneously. So, it's important to pay attention to which type of graph you're analyzing and the different meanings of each.

On a position vs. time graph, the $y$-value represents *position* and the slope represents *velocity*.

On a velocity vs. time graph, the $y$-value represents *velocity* and the slope represents *acceleration*. The area under the curve represents *displacement*.

On an acceleration vs. time graph, the $y$-value represents *acceleration*. The area under the curve represents *change in velocity*.

Keeping careful track of units, both on the $y$-axis and in calculations, can help you remember what quantity you're dealing with.