Section 1.5: Translating Between Representations
Physics problems can often be described in multiple ways — words, equations, diagrams, tables, or graphs. Translating between these representations strengthens understanding and allows us to solve problems more flexibly.
Common Representations:
- Verbal description: A story of motion or forces.
- Equations: Mathematical form of the situation.
- Graphs: Position-time, velocity-time, or acceleration-time graphs.
- Diagrams: Motion diagrams, free-body diagrams.
- Tables: Values at discrete times.
Why Translate?
- Each form highlights different aspects (e.g., slope on a position-time graph is velocity).
- Translating checks for consistency across forms.
- Some problems are easier to solve in one representation than another.
Example 1: Motion Graphs
A car starts from rest and accelerates at \(2\ \text{m/s}^2\). Translate this description into an equation, a position-time graph, and a velocity-time graph.
- Equation: \(x = \tfrac{1}{2} (2) t^2 = t^2\).
- Velocity equation: \(v = 0 + 2t\).
- Graphs: Position-time is a parabola opening upward; velocity-time is a straight line with slope 2.
Example 2: Verbal to Algebraic
A ball is thrown upward with initial velocity \(10\ \text{m/s}\). Gravity is \(9.8\ \text{m/s}^2\) downward. Write the equations of motion and sketch velocity-time graph.
\( v(t) = 10 - 9.8t \).
\( y(t) = 10t - 4.9t^2 \).
The velocity-time graph is a straight line with negative slope crossing zero at about 1.02 s (the peak).
Practice Problems
- A runner moves with constant speed of \(5\ \text{m/s}\). Represent this with a table of values, an equation, and a position-time graph.
- Sketch velocity-time and acceleration-time graphs for a car uniformly decelerating from \(20\ \text{m/s}\) to rest in 5 s.
- A motion diagram shows equal spacing of dots every second. Translate this into a verbal description and position-time graph.
- Given the equation \(x = 4t - t^2\), describe the motion in words and sketch a velocity-time graph.