Position-Time Graphs - Physics

1. Fundamental Concepts

Definition: A position-time graph represents the motion of an object by plotting its position against time.
Slope Interpretation: The slope of a position-time graph at any point gives the velocity of the object at that instant.
Constant Velocity: A straight line on a position-time graph indicates constant velocity.

2. Key Concepts

Basic Rule: $x(t) = x_0 + vt$

Slope Calculation: The slope is calculated as \frac{\Delta x}{\Delta t}

Application: Used to analyze and predict motion in physics

3. Examples

Example 1 (Basic)

Problem: Given the position-time graph where an object starts at $x_0 = 2$ meters and moves with a constant velocity of $v = 3$ m/s, find the position at $t = 4$ seconds.

Step-by-Step Solution:

Use the equation $x(t) = x_0 + vt$.
Substitute the given values: $x(4) = 2 + 3 \cdot 4$.
Calculate: $x(4) = 2 + 12 = 14$ meters.

Validation: Substitute $t = 4$ into the equation → Original: $2 + 3 \cdot 4 = 14$; Simplified: $14$ meters ✓

Example 2 (Intermediate)

Problem: An object has a position-time graph described by the equation $x(t) = 5t^2 - 2t + 1$. Find the velocity at $t = 2$ seconds.

Step-by-Step Solution:

Find the derivative of $x(t)$ to get the velocity function $v(t)$: $v(t) = \frac{dx}{dt} = 10t - 2$.
Substitute $t = 2$ into $v(t)$: $v(2) = 10 \cdot 2 - 2$.
Calculate: $v(2) = 20 - 2 = 18$ m/s.

Validation: Substitute $t = 2$ into the velocity equation → Original: $10 \cdot 2 - 2 = 18$; Simplified: $18$ m/s ✓

4. Problem-Solving Techniques

Graphical Analysis: Use graphs to visualize the relationship between position and time.
Derivative Application: Calculate derivatives to find velocities from position functions.
Equation Substitution: Substitute known values directly into equations for quick solutions.