A forklift drives 12 m forward, 4 m backward, then 2 m forward. Which pair states the distance traveled and the displacement?
A city bus sets off and holds a steady 8 m/s along a straight street for 3 s. Then the driver sees the next stop and brakes steadily, reaching rest at t = 7 s. What is the bus's displacement over those 7 s? How far down the street does it end up?
You've already seen that the area between the line and the time axis is the displacement, and why. Now you calculate that area, for graphs built from straight segments.
Read each shape off the graph. A shape's width runs along the time axis, in s. Its height runs from the axis out to the line, in m/s. The bus's region splits at t = 3 s into one rectangle and one triangle.
The graph shows the bus's velocity: steady at 8 m/s for 3 s, then falling steadily to rest at t = 7 s. What is the bus's displacement over the whole 7 s?
Split the region at t = 3 s:
a rectangle, then a triangle
Read each shape off the graph:
rectangle: width 3 s, height 8 m/s
triangle: base 4 s, height 8 m/s
Find each piece's area:
rectangle: 8 × 3 = 24 m
triangle: ½ × 4 × 8 = 16 m
Give each piece its sign:
both pieces sit above the axis, so both count positive
Add the signed pieces:
Δx = 24 + 16
Δx = 40 m
The graph shows a monorail's velocity, steady from t = 2 s to t = 6 s. Calculate the monorail's displacement between t = 2 s and t = 6 s. Enter the value in meters, sign included.
The graph shows a glider's velocity rising steadily from zero at t = 2 s. Calculate the glider's displacement between t = 2 s and t = 8 s. Enter the value in meters, sign included.
Below the axis the routine is the same, with one extra care. Read every height as a positive length, from the axis down to the line. Then apply the minus sign as its own step, one sign per piece.
The graph shows a diver's underwater scooter: steady at −6 m/s for 4 s, then rising steadily to rest at t = 7 s. What is the scooter's displacement over the whole 7 s?
Split the region at t = 4 s:
a rectangle, then a triangle
Read each shape off the graph, heights as positive lengths:
rectangle: width 4 s, height 6 m/s, below the axis
triangle: base 3 s, height 6 m/s, below the axis
Find each piece's area:
rectangle: 6 × 4 = 24 m
triangle: ½ × 3 × 6 = 9 m
Give each piece its sign:
both pieces sit below the axis: −24 m and −9 m
Add the signed pieces:
Δx = (−24) + (−9)
Δx = −33 m
The graph shows a tugboat's velocity, steady and below the time axis from t = 1 s to t = 7 s. Calculate the tugboat's displacement between t = 1 s and t = 7 s. Enter the value in meters, sign included.
The same split, sign, and add routine handles any velocity-time graph made of straight segments.
The graph shows a delivery trike's velocity: steady at 5 m/s for 3 s, then falling steadily to zero at t = 7 s. Calculate the trike's displacement over the whole 7 s. Enter the value in meters, sign included.
The graph shows a freight train's velocity over 8 s, crossing the time axis at t = 4 s. What is the train's displacement over the whole 8 s?
The graph shows a luggage cart's velocity: falling steadily from zero at t = 1 s to -4 m/s at t = 9 s. Calculate the cart's displacement between t = 1 s and t = 9 s. Enter the value in meters, sign included.