A-level Physics (Advancing Physics)/Graphs

From Wikibooks, open books for an open world
Jump to navigation Jump to search

There are two types of graphs of motion you need to be able to use and understand: distance-time graphs and velocity-time graphs.

Distance-time Graphs[edit]

An object travels at a constant rate for 6 seconds, stops for 5 seconds, returns to its original position in the next 7 seconds, travelling more slowly in the middle section of its return journey.

A distance-time graph plots the distance of an object away from a certain point, with time on the x-axis and distance on the y-axis.There are several types of graphs of motion you need to be able to use and understand: distance-time graphs, position-time graphs, and velocity-time graphs.

Position-time Graphs or Displacement - Time Graphs[edit]

Distance-Time Graphs give you speed, but speed is never negative so you can only have a positive slope in a distance-time graph. Position-Time graphs show displacement, have direction, and from which you can calculate velocity. If we were to imagine the line on the position-time graph to the right as a function f(t), giving an equation for s = f(t), we could differentiate this to gain:

,

where s is displacement, and t is time. By finding f'(t) at any given time t, we can find the rate at which distance is increasing (or decreasing) with respect to t. This is the gradient of the line. A positive gradient means that distance is increasing, and a negative gradient means that distance is decreasing. A gradient of 0 means that the object is stationary. The velocity of the object is the rate of change of its displacement, which is the same as the gradient of the line on a distance-time graph. This is not necessarily the same as the average velocity of the object v:

Here, t and s are the total changes in displacement and time over a certain period - they do not tell you exactly what was happening at any given point in time.

Velocity-time Graphs[edit]

An object accelerates for 6 seconds, hits something, accelerates for 5 seconds and then decelerates to a stop in the remaining 6.5 seconds of its journey.

A velocity-time graph plots the velocity of an object, relative to a certain point, with time on the x-axis and velocity on the y-axis. We already know that velocity is the gradient (derivative) of the distance function. Since integration is the inverse process to differentiation, if we have a velocity-time graph and wish to know the distance travelled between two points in time, we can find the area under the graph between those two points in time. In general:

If

where v is velocity (in ms−1), t is time (in s), and s is the distance travelled (in m) between two points in time t1 and t2.

Also, by differentiation, we know that the gradient (or derivative of v = f(t)) is equal to the acceleration of the object at any given point in time (in ms−2) since:

Questions[edit]

1. In the following distance-time graph, what is the velocity 4 seconds after the beginning of the object's journey? Distance-time graph example.png

2a)Describe this person's movements.

2b)What is the velocity at 12 seconds?

3. In the following velocity-time graph, how far does the object travel between 7 and 9 seconds? Velocity-time graph example.png

4. What is the object's acceleration at 8 seconds?

5. A car travels at 10ms−1 for 5 minutes in a straight line, and then returns to its original location over the next 4 minutes, travelling at a constant velocity. Draw a distance-time graph showing the distance the car has travelled from its original location.

6. Draw the velocity-time graph for the above situation.

The following question is more difficult than anything you will be given, but have a go anyway:

7. The velocity of a ball is related to the time since it was thrown by the equation . How far has the ball travelled after 2 seconds?

Worked Solutions