2.6 Graphical Analysis of Motion

2.6.1 Distance-Time Graph

A distance-time graph shows the relationship between distance (s) and time (t). Time is plotted on the x-axis, and distance on the y-axis.

  • The graph is always in the positive X plane.
  • The gradient (slope) of the graph gives speed.

Calculating Gradient:

  1. Choose two points P₁ and P₂
  2. Determine coordinates P₁(t₁, s₁) and P₂(t₂, s₂)
  3. Calculate Δt = t₂ - t₁ and Δs = s₂ - s₁
  4. Gradient = Δs / Δt = speed

Example 2.5: Peshawar to Islamabad through M1

A car travels from Peshawar to Islamabad, stopping for 30 minutes at a rest area.

Results:

  • Peshawar to rest area: 100 km/hr (27.78 m/s)
  • Rest area to Islamabad: 50 km/hr (13.89 m/s)
  • Overall journey: 70 km/hr (19.44 m/s)

2.6.2 Speed-Time Graph

A speed-time graph shows the relationship between speed (v) and time (t). Speed is plotted on the y-axis, and time on the x-axis.

  • The slope of the graph gives the magnitude of acceleration.
  • The area under the graph gives the distance traveled.

Calculating Area:

Rectangle: Area = v × t

Triangle: Area = 0.5 × v × t

Example 2.6: Graphical Representation of Speed of Car

A car increases speed from 0 to 30 m/s in 20s, maintains speed for 30s, then decreases to 0 in 10s.

Results:

  • Acceleration (0-20s): 1.5 m/s²
  • Acceleration (20-50s): 0 m/s²
  • Acceleration (50-60s): -3 m/s²
  • Total distance: 1350 m
  • Average speed: 22.5 m/s