Motion

##Position

[[☃ image 2]]

This man's car has broken down. To describe the car’s location to a tow truck driver, the man needs to give a reference point like an intersection. Then the car's position can be described with two additional pieces of information:

• the car’s straight-line distance from the reference point
• the car’s direction from the reference point

But what if the car’s position is changing over time, like the car in the image below? To describe a car’s motion, we need additional terms.

[[☃ explanation 1]]

Speed and velocity

[[☃ image 7]]

An object’s speed is a measure of how fast it’s moving. It's calculated by measuring the distance the object travels and dividing by the amount of time it takes to travel that distance:

$\text{speed}=\dfrac{\text{distance}}{\text{time}}$

If an object’s speed is changing throughout the distance measured, like a car speeding up as it gets on the highway, then this equation gives us its average speed during that time.

Velocity is also a measure of how fast an object moves. But velocity isn’t exactly the same as speed because velocity also gives the direction the object moves.

The units for speed and velocity can be any distance unit divided by any time unit. Common everyday units are miles per hour $\left(\tfrac{\text{mi}}{\text{h}}\right)$ and kilometers per hour $\left(\tfrac{\text{km}}{\text{h}}\right).$ The scientific units for speed and velocity are meters per second $\left(\tfrac{\text{m}}{\text{s}}\right).$

Distance-time graphs

A distance vs. time graph models the amount of time an object travels on the [[☃ definition 3]]
and the distance it travels in that time on the [[☃ definition 1]].

This table shows different amounts of time in hours $(\text{hr})$ and the distance the car has travelled in miles $(\text{mi})$ after that amount of time.

[[☃ graded-group-set 6]]

The graph lets us see a pattern in the data. Imagine connecting the dots. What shape would be formed?

[[☃ graded-group-set 2]]

Adding a line to the graph helps us visualize its steepness, or slope. The slope is the rate at which the $y$-axis variable changes compared to the $x$-axis variable.

[[☃ graded-group-set 4]]

This can be written as an equation:

$\text{slope}=\dfrac{\text{distance}}{\text{time}}$

We learned:

$\text{speed}=\dfrac{\text{distance}}{\text{time}}$

Therefore, the slope of the graphed line equals the car’s speed. This is true for any distance vs. time graph, not just for our car.

[[☃ graded-group-set 5]]

[[☃ graded-group-set 3]]

##Acceleration

If an object’s velocity is changing, it is accelerating. Velocity includes an object’s speed and its direction of motion. So, an object is accelerating if its speed is changing, direction is changing, or both are changing.

[[☃ graded-group-set 1]]

Acceleration is a measure of how quickly an object’s velocity changes. All cars on the highway can change from a speed of $0$ to a speed of $60,\tfrac{\text{mi}}{\text{hr}}.$ But if one car speeds up to $60,\tfrac{\text{mi}}{\text{hr}}$ more quickly (in less time) than another, then it has a greater acceleration.

##Communicating information about motion

Suppose you’re reading a book in a moving car. Compared to you and everything else in the car, the book's not changing position. In the reference frame of the car, it's not moving. But if your friends standing on the sidewalk see the book through the window, they’ll see that it is moving compared with the ground.

Both of these views of the book's motion are valid. But to communicate how the book is moving, the reference frame—what the motion’s being compared to—must be clear.

When communicating amounts, called quantities, the units must also be clear. Below are some quantities with their units in red:

• A skateboarder travels a distance of $20,\text{\red{m}}.$
• It takes her $4,\text{\red{s}}.$
• Her average speed is $5,\red{\tfrac{\text{m}}{\text{s}}}.$

In graphs and data tables, each labeled quantity is followed by its units in parentheses $(,,)$ so the units don’t need to be written after every number. A data table and graph of the skateboarder's motion are shown below with the units in red:

[[☃ interactive-graph 1]]

Misunderstandings can happen when units aren’t clear. In 1998 NASA’s Mars Climate Orbiter mission launched to study the Martian atmosphere and surface. But some of the commands had been entered using English units and some using metric. The result? The mission failed when the orbiter burned up in Mars’s atmosphere.

[[☃ image 1]]

##Key points

• Average speed measures how fast an object moves.

• $\text{speed}=\dfrac{\text{distance}}{\text{time}}$

• Velocity measures how fast and in what direction an object moves.

• The slope of a line on a distance vs. time graph equals average speed.

• Acceleration measures how quickly an object changes velocity.
  An object is accelerating if its speed, direction, or both are changing.

• Information about an object’s motion must include the reference frame and units used.