(455) Accelerated motion – graphing [IB Physics SL/HL]

Overview of Accelerated Motion

  • Understanding motion includes analyzing graphs of displacement, velocity, and acceleration.

Key Concepts

  • Velocity: Defined as distance over time.

  • Acceleration: The second derivative of position with respect to time; the third derivative is called jerk, represented humorously as "Don't be a jerk."

Example Experiment: Rolling a Can of Soup

  • Setup: A can of soup is rolled up an incline with a Sonic Ranger to measure displacement over time.

  • Displacement: Initially zero at the start, reaching a maximum height, then returning down.

Graphs of Motion

Displacement vs. Time

  • Graph is a smooth curve (parabola) that starts from zero, peaks, and returns to zero.

Velocity vs. Time

  • Velocity graph is derived from the displacement graph:

    • The gradient (slope) represents velocity.

    • At peak displacement, the tangent is flat (slope = 0), indicating zero velocity.

    • As displacement decreases, velocity becomes negative as the can rolls back down.

Acceleration vs. Time

  • Acceleration graph is derived from the velocity graph:

    • The gradient of the velocity graph represents acceleration.

    • At points of zero velocity (flat tangent), acceleration is also zero.

    • Negative acceleration occurs as the can rolls downhill, consistent with the negative gradient.

The Trick for Graphing Relationships

  • Down, Down: To move from displacement to velocity and then from velocity to acceleration, take the gradient (derivative).

  • Up: To move from velocity back to displacement, take the area under the curve (integral).

Example Calculation

Area Under the Velocity Curve

  • For distance traveled from a velocity graph:

    • Calculate areas of geometric shapes under the curve:

      • Triangle: Base of 1 sec, height of 5 m/s

        • Area = (1 * 5) / 2 = 2.5 m

      • Rectangle: Width of 2 sec (from 1 to 3), height of 5 m/s

        • Area = 2 * 5 = 10 m

  • Total Distance = 2.5 m + 10 m = 12.5 m

Conclusion

  • Understanding the relationships between displacement, velocity, and acceleration through their graphs enhances the comprehension of motion and can aid in solving physics problems effectively.