Section 1.7: Graphical Analysis

Graphical analysis helps understand motion by studying displacement-time, velocity-time, and acceleration-time graphs. Important insights like average speed, total distance, and acceleration patterns can be extracted from graphs.

Uniform Velocity:

Displacement-time graph is a straight line. Slope = velocity.

Displacement-Time Graph (Uniform Velocity)
Non-uniform Velocity:

Displacement-time graph is curved. Instantaneous velocity = slope at a point.

Displacement-Time Graph (Non-uniform Velocity)
Uniform Acceleration:

Velocity-time graph is a straight line. Slope = acceleration. Area under graph = displacement.

Velocity-Time Graph (Uniform Acceleration)
Non-uniform Acceleration:

Velocity-time graph is curved. Slope = instantaneous acceleration. Area under graph = displacement.

Velocity-Time Graph (Non-uniform Acceleration)
Important Concepts:
  • Area under velocity-time graph: Total displacement.
  • Total distance: Sum of absolute areas (upward and downward segments).
  • Average speed: Total distance ÷ total time.

Example 1

A car moves with uniform acceleration. The velocity-time graph is a straight line from 0 to 20 m/s in 5 s. Find displacement using area under the graph.

Displacement = area of triangle = 0.5 × base × height = 0.5 × 5 × 20 = 50 m

Example 2

The velocity-time graph of a particle is curved. Between t=0 and t=4 s, approximate area under the curve using trapezoids to find displacement.

This requires adding areas of trapezoids or using integration if function is known. Approximate numerical value depends on graph data.

Practice Problems

  1. A car moves at uniform velocity of 15 m/s for 10 s. Draw displacement-time graph and find displacement.
  2. Velocity-time graph shows a straight line from 0 to 25 m/s over 8 s. Find acceleration and displacement.
  3. A particle moves such that displacement-time graph is a parabola. Describe motion and calculate instantaneous velocity at t=3 s.
  4. Given a velocity-time graph that slopes downward from 20 m/s to 0 in 5 s, find total displacement and average speed.
  5. A particle’s velocity-time graph consists of two segments: 0→10 m/s in 4 s, then constant 10 m/s for 3 s. Find total distance.
  6. Sketch a non-uniform acceleration graph and explain how to find instantaneous acceleration at t=2 s.
  7. A cyclist moves with velocity given by v(t)=2t m/s. Draw velocity-time graph and find displacement in 5 s.
  8. Given a piecewise velocity-time graph with positive and negative segments, determine total distance and displacement.
  9. Analyze a car’s motion using given v-t graph and calculate acceleration during first 3 s.
  10. Construct displacement-time graph from given velocity-time data and determine slope at t=4 s.
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