2.3 Properties of Motion Graphs

Slope & Area of Motion Graphs

• Three types of graph that are used to represent motion are displacement-time graphs, velocity-time graphs and acceleration-time graphs

• To interpret these find either the slope or the area 

• Slope is found using , which can be related to any similar algebraic relationship, for example, 
  • The slope of a distance-time graph will equal velocity

• Area is found using area = change in y × change in x, which can be related to any similar algebraic relationship, for example, velocity  × time = distance
  • The area under a velocity-time graph will equal distance

Finding Slope of a Straight Line

• The slope of a line is found using the formula:

slope =

• This is also written:

slope = 

• To find the slope;
  • Identify two points as far apart as possible which intercept with the grid of the graph paper

• • The points must be on the line itself, not the furthest apart plots
  • Clearly mark the points chosen
  • Find the co-ordinates of the two points
  • Calculate to find the slope

= 3.125

• Use the values on the axes of the graph (or in the data) to identify the correct significant figures
• Use the units to find the correct units, which also equal 

units of slope =

• On this graph all values are to 2 significant figures, 
• Therefore slope = 3.1 m s⁻¹

Finding the Area Under the Line

• The area underneath a line represents the x axis multiplied by the y axis.
• For example, if velocity is plotted on the y-axis and time is on the x-axis, then
  • area = velocity × time = distance
• When the area is a rectangle or a triangle it is easy to find by calculating

area of rectangle = base × height

area of a triangle = ½ base × height

To find the area under a straight-line graph treat is as any rectangle or triangle

• When the line is curved, area is found though the following steps
  • Divide the shape into rectangles and triangles as shown
  • Find the area for each by calculating
  • Count the remaining squares
  • Add the totals together

Find area under a curve by dividing it into sections which reduce the number of squares to be counted to the minimum

Non-uniform Acceleration

• In certain cases, acceleration changes throughout a period of motion
• An example of this is a skydiver falling due to their weight, against air resistance
  • As speed increases, air resistance increases 
  • The increasing resultant force against the direction of motion acts to reduce acceleration
  • Eventually the weight and air resistance balance and acceleration become zero
  • Acceleration for a falling object is conventionally shown as a negative value

• Non-uniform acceleration will produce a curved velocity-time graph

Finding the Slope of a Curved Line

• Some questions ask for velocity or acceleration at a particular point or time
• This is called 'instantaneous' velocity or acceleration
• It is found by taking the slope of the curved line

• Identify the point on the x-axis which corresponds with the question
  • Draw a line vertically to meet the curve
• Take a tangent to the curve at this point
  • Do this carefully, trying out more than one attempt before choosing the correct tangent

Tangents are a useful way to find the slope of a curved line

• Draw the tangent in place, making it as long as will fit onto the graph
  • Follow the steps above to find the slope of the tangent

Worked Example

Step 1: Draw a tangent to the curve at the point where t = 5 s

Step 2: Select points on the graph which are very far apart, and determine the corresponding values of x and y, write these down

Increase in y = (58 -20) = 38 m s⁻¹

Increase in x = (9.75 - 0.75) = 9.0 s

Step 3: Calculate the slope

slope =  = 4.222  m s⁻²

Step 4: Write the answer to the correct significant figures

The acceleration at 5.0 s = 4.2 m s⁻² (2 sf)

Exam Tip