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Oct 21, 2023 · Solve for speed, distance, time and rate with formulas s=d/t, d=st, d=rt, t=d/s. Calculate rate of speed given distance and time. Find mph, miles per hour, km/hour.
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This formula shows that: Speed increases if you cover more distance in the same amount of time, or if you cover a distance in less time. Distance can be calculated if you know the speed and time, with the formula: distance = speed × time.
Speed is a compound measure of distance divided by time. Speed units are read as length unit per time unit. Units of speed include miles per hour (mph), kilometres per hour (km/h or kmph) and...
- Overview
- How are position vs. time graphs useful?
- What does the vertical axis represent on a position graph?
- What does the slope represent on a position graph?
- What does the curvature on a position graph mean?
- Example 1: Hungry walrus
- Example 2: Happy bird
See what we can learn from graphs that relate position and time.
How are position vs. time graphs useful?
Many people feel about graphs the same way they do about going to the dentist: a vague sense of anxiety and a strong desire for the experience to be over with as quickly as possible. But position graphs can be beautiful, and they are an efficient way of visually representing a vast amount of information about the motion of an object in a conveniently small space.
What does the vertical axis represent on a position graph?
The vertical axis represents the position of the object. For example, if you read the value of the graph below at a particular time you will get the position of the object in meters.
Try sliding the dot horizontally on the graph below to choose different times and see how the position changes.
Many people feel about graphs the same way they do about going to the dentist: a vague sense of anxiety and a strong desire for the experience to be over with as quickly as possible. But position graphs can be beautiful, and they are an efficient way of visually representing a vast amount of information about the motion of an object in a convenient...
The vertical axis represents the position of the object. For example, if you read the value of the graph below at a particular time you will get the position of the object in meters.
Try sliding the dot horizontally on the graph below to choose different times and see how the position changes.
Concept check: What is the position of the object at time t=5 seconds according to the graph above?
The slope of a position graph represents the velocity of the object. So the value of the slope at a particular time represents the velocity of the object at that instant.
To see why, consider the slope of the position vs. time graph shown below:
[Wait, why is the vertical axis called x?]
The slope of this position graph is slope=riserun=x2−x1t2−t1 .
This expression for slope is the same as the definition of velocity: v=ΔxΔt=x2−x1t2−t1 . So the slope of a position graph has to equal the velocity.
This is also true for a position graph where the slope is changing. For the example graph of position vs. time below, the red line shows you the slope at a particular time. Try sliding the dot below horizontally to see what the slope of the graph looks like for particular moments in time.
Look at the graph below. It looks curvy since it's not just made out of straight line segments. If a position graph is curved, the slope will be changing, which also means the velocity is changing. Changing velocity implies acceleration. So, curvature in a graph means the object is accelerating, changing velocity/slope.
On the graph below, try sliding the dot horizontally to watch the slope change. The first hump between 1 s and 5 s represents negative acceleration since the slope goes from positive to negative. For the second hump between 7 s and 11 s , the acceleration is positive since the slope goes from negative to positive.
Concept check: What is the acceleration of the object at t=6 s according to the graph above?
[Show me the answer.]
To summarize, if the curvature of the position graph looks like an upside down bowl, the acceleration will be negative. If the curvature looks like a right side up bowl, the acceleration will be positive. Here's a way to remember it: if your bowl is upside down all your food will fall out and that is negative. If your bowl is right side up, all your food will stay in it and that is positive.
Finding the velocity at 2 s :
We can find the velocity of the walrus at t=2 s by finding the slope of the graph at t=2 s : slope=x2−x1t2−t1(use the formula for slope) Now we will pick two points along the line we are considering that conveniently lie at a hashmark so we can determine the value of the graph at those points. We'll choose the points (0 s,1 m) and (4 s,3 m) , but we could pick any two points between 0 s and 4 s . We must plug in the later point in time as point 2, and the earlier point in time as point 1. slope=3 m−1 m4 s−0 s(Pick two points and plug the x values into the numerator and the t values into the denominator.) slope=2 m4 s=12 m/s(Calculate and celebrate.) So, the velocity of the walrus at 2 s was 0.5 m/s .
Finding the velocity at 5 s :
To find the velocity at 5 s , we just have to note that the graph is horizontal there. Since the graph is horizontal, the slope is equal to zero, which means that the velocity of the walrus at 5 s was 0 m/s .
Finding the velocity at 8 s :
slope=x2−x1t2−t1(Use the formula for slope.) We'll pick the points at the beginning and end of the final line segment, which are (6 s,3 m) and (9 s,0 m) . slope=0 m−3 m9 s−6 s(Pick two points and plug the x values into the numerator and the t values into the denominator.) slope=−3 m3 s=−1 m/s(Calculate and celebrate.) So, the velocity of the walrus at 8 s was −1 m/s .
The motion of an extraordinarily jubilant bird flying straight up and down is given by the graph below, which shows the vertical position y as a function of time t . Answer the following questions about the motion of the bird.
What was the average velocity of the bird between t=0 s and t=10 s ?
Calculate the total distance travelled by the object - its motion is represented by the velocity-time graph below. Here, the distance travelled can be found by calculating the total area of the...
Convert a journey distance in kilometres and the time taken in hours, to an average speed in mph. Determine the average speed of a runner in mph, from a track run distance in metres, and the stopwatch time in minutes or seconds.
As discussed in a previous part of Lesson 4, the shape of a velocity vs. time graph reveals pertinent information about an object's acceleration. For example, if the acceleration is zero, then the velocity-time graph is a horizontal line - having a slope of zero.
