1. Working Scientifically:
Gradients
on graphs

2. How to calculate the gradient (slope) of a straight-line graph…
Learning Objectives
You should learn :
How to calculate the gradient (slope) of a straight-line graph…
…and a curved graph
About the equations for different straight-line graphs.
The meaning of gradients on graphs for distance‒time and velocity‒time.

3. …the gradient of a graph…?
Looking at graphs
What do we mean by…
…the gradient of a graph…?

4. Looking at graphs Where is the curve steepest?
Where does he accelerate the most?
Where is the steepest gradient?

5. Looking at graphs Here the gradient is higher, and
the rate of change of speed is higher.

6. Looking at graphs Here the gradient is smaller, and
the rate of change of speed is smaller.

7. …how do you calculate the gradient of it?
Gradient of a graph
The slope or gradient of a graph can give you important information.
…how do you calculate the gradient of it?
But when you’ve drawn a graph…
It can be done in 3 steps:

8. Gradient of a graph
Here is the graph from an experiment to measure the extension of a spring (see page 66)
It shows the load applied to the spring (in N) and the extension it causes (in mm).

9. Gradient of a graph Step 1 Draw a large right-angled triangle
as shown here:

10. Gradient of a graph Step 2 Find the value of the two sides Y and X
in the units of
the graph.
In the diagram,
Y = 20 mm and X = 60 N. Can you see why?

11. gradient = Y = 20 mm Gradient of a graph Step 3
Calculate the gradient (slope) of the graph
by dividing Y/ X
(Keep the units with the numbers, so that you find the unit of
the gradient.) X
From the diagram:
gradient = Y 
60 N
= 20 mm
= 0.33 mm/N

12. gradient = Δy = 16 - 4 m/s = 12 m/s What if the graph is curved ?
First, add a tangent at the point P where you want to find the gradient:
Then find the gradient of that tangent line. Δx
gradient = Δy  s
= m/s
19 s
= 12 m/s
= 0.63 m/s2

13. Equations for straight-line graphs
Let’s look at 3 examples of straight-line graphs,
together with their algebraic equations.
These are the 3 graphs we’ll look at:

14. Directly proportional
Equations for straight-line graphs
Directly proportional
For an equation like
Y = k X
the graph is
a straight line,
through the origin.
The gradient (slope) = k 
For an example, see Hooke’s Law .

15. Linear but not directly proportional
Equations for straight-line graphs
Linear but not directly proportional
For an equation like
Y = k X + c
the graph is
a straight line, but
not through the origin.
Like before, the gradient = k
The intercept gives you the value of c.
For an example see v = a t + u.

16. Inversely proportional
Equations for straight-line graphs
Inversely proportional
If pressure P and
volume V are
inversely proportional
a graph of P against V
would give a curve.
The equation is P = k × 1 V
so to get a straight line, 1 V
we have to plot: P against

17. Looking at graphs of movement
Let’s look at some examples of graphs with different gradients…

18. Distance – time graphs
Example 1

19. Distance – time graph The graph is flat. What is happening here?
The distance is not changing.
The gradient is zero.
The speed is zero.

20. Distance – time graphs
Example 2

21. Distance – time graph The distance is changing.
What is happening here?
The object is moving.
The constant gradient means a constant speed.

22. Distance – time graph What do the numbers tell you?
The object has gone 20 m in 2 seconds.
What is its speed?

23. Distance – time graph Speed = = = 10 m/s distance travelled 20 m
time taken =
20 m
2 s
= 10 m/s

24. Distance – time graphs
Example 3

25. Distance – time graph What is happening here?
The gradient (slope) is increasing.
So the speed is increasing.
The object is accelerating.

26. Distance – time graphs
Example 4

27. Distance – time graph When is it travelling fastest?
What is happening to the lift?
At B. (steepest)
When is it stationary?
At A and at C. (flat)

28. Distance – time graph When is it accelerating? Between A-B.
When is it decelerating?
Between B-C.

29. Distance – time graph How far does it travel? 35 metres
How long does it take?
7 seconds
What is its average speed?
5 m/s

30. …graphs of velocity) against time
Now let’s look at a different set of graphs…
…graphs of velocity)
against time

31. Velocity – time graph
Example 1

32. Velocity – time graph The graph is flat. What is happening here?
The velocity is not changing.
It is not accelerating or decelerating.

33. Velocity – time graph
Example 2

34. Velocity – time graph What is happening here?
The velocity is changing.
The object is accelerating.

35. Velocity – time graph
Example 3

36. Velocity – time graph Now the gradient is steeper.
The velocity is changing more quickly.
The object has a greater acceleration.

37. Velocity – time graph
Example 4

38. Velocity – time graph What is happening to this car?
When is it travelling fastest?
Between C and D.
When is it stationary?
At A and at E.

39. Velocity – time graph When is it accelerating? Between A and C.
When is it decelerating?
Between D and E.

40. Velocity – time graph
Example 5

41. Velocity – time graph What is happening to this lift?
When is it travelling fastest?
At B.
When is it stationary?
At A and at C.

42. Velocity – time graph When is it accelerating? Between A and B.
When is it decelerating?
Between B and C.

43. Velocity – time graph
What is the acceleration between A and B ?

44. Acceleration = gradient = Δy = 10 m/s
Velocity – time graph Δx
Acceleration = gradient = Δy 
5 s
= 10 m/s
= 2 m/s2

45. Velocity – time graph
What is the deceleration between B and C ?

46. Deceleration = gradient = Δy = 10 m/s
Velocity – time graph Δx
Deceleration = gradient = Δy 
2 s
= 10 m/s
= 5 m/s2
Or acceleration = ‒5 m/s2

47. Drawing a conclusion
Every experiment has a ‘conclusion'. This is a summary of what you found out(or sometimes what you didn’t find).
Always look at your results or graph or chart to decide what you have discovered.
What reliable and valid deduction can be made from your results? What pattern can you see?
If possible, try to use your scientific knowledge to explain and justify your conclusion.

48. Evaluating your evidence
As part of the conclusion, comment should be made on the reliability and validity of the measurements that are made.
The following can improve the reliability of the data:
Taking more measurements, perhaps changing the range.
Improving the design of the investigation.
Checking the results by an alternative method.
Looking up data from secondary sources(eg. Books or internet)
Seeing if other people following your method get the same results(i.e are your results reproducible)