Motion in One Dimension: Position, Velocity, and Acceleration

Motion in One Dimension

Position-Time Graphs & Velocity

Position-time graphs are fundamental tools for visualizing and analyzing the motion of objects in one dimension. The vertical axis (y-axis) represents position (x, in meters), while the horizontal axis (x-axis) represents time (t, in seconds). These graphs allow us to determine how an object's position changes over time and to extract information about its velocity.

• Position-Time Graph: Shows how an object's position changes with time.

• Velocity (Average): Defined as the change in position divided by the change in time: $v_{avg} = \frac{\Delta x}{\Delta t}$

• Graphical Interpretation: The slope of the position-time graph at any interval gives the average velocity over that interval.

• Upward Slope: Object is moving forward (positive velocity).

• Horizontal/Flat Slope: Object is at rest (velocity is zero).

• Downward Slope: Object is moving backward (negative velocity).

• Steeper Slope: Indicates a higher magnitude of velocity.

• Flatter Slope: Indicates a lower magnitude of velocity.

Example: If you walk 6 m forward in 3 s, stop for 1 s, then run 6 m back in 1 s, the position-time graph would show segments with different slopes corresponding to each phase of motion.

Curved Position-Time Graphs & Acceleration

When the position-time graph is curved (not a straight line), the object's velocity is changing, indicating acceleration is present.

• Curving Up (Concave Up): Positive acceleration (object is speeding up in the positive direction).

• Curving Down (Concave Down): Negative acceleration (object is slowing down or speeding up in the negative direction).

• Straight Line: Constant velocity, zero acceleration.

Instantaneous Velocity

There are two types of velocity to consider:

• Average Velocity: Slope between two points on the position-time graph.

• Instantaneous Velocity: Slope of the tangent line at a single point on the position-time graph.

At the top or bottom of a position graph (maximum or minimum), the instantaneous velocity is zero. The tangent line at a point gives the instantaneous velocity at that moment.

Velocity-Time Graphs & Acceleration

Velocity-time graphs plot velocity (y-axis) versus time (x-axis). The slope of the velocity-time graph gives the acceleration:

• Acceleration (Average): $a_{avg} = \frac{\Delta v}{\Delta t}$

• Graphical Interpretation: The slope of the velocity-time graph at any interval gives the average acceleration.

• Steeper Slope: Higher magnitude of acceleration.

Calculating Displacement from Velocity-Time Graphs

The area under the curve of a velocity-time graph between two points gives the displacement ($\Delta x$):

• Area above the time axis: Positive displacement

• Area below the time axis: Negative displacement

• For rectangles: $A = b \times h$

• For triangles: $A = \frac{1}{2} b \times h$

Equations of Motion (Kinematics Equations)

For constant acceleration, the following equations (Uniformly Accelerated Motion, UAM) are used:

• 1. $v = v_0 + at$

• 2. $v^2 = v_0^2 + 2a\Delta x$

• 3. $x = x_0 + v_0 t + \frac{1}{2} a t^2$ or $\Delta x = v_0 t + \frac{1}{2} a t^2$

• 4. $\Delta x = \left(\frac{v_0 + v}{2}\right)t$

To solve motion problems, identify the known and unknown variables, select the appropriate equation, and solve for the target variable.

Vertical Motion & Free Fall

Objects in free fall experience constant acceleration due to gravity ($g = 9.8\,\mathrm{m/s^2}$ downward). The same kinematics equations apply, with $a_y = -g$ if upward is positive.

• Vertical UAM Equations (for free fall):

• 1. $v_y = v_{0y} + a_y t$

• 2. $v_y^2 = v_{0y}^2 + 2a_y \Delta y$

• 3. $\Delta y = v_{0y} t + \frac{1}{2} a_y t^2$

• 4. $\Delta y = \left(\frac{v_{0y} + v_y}{2}\right)t$

Catch Up or Overtake Problems

When one object catches up to another, they are at the same position at the same time. Set their position equations equal and solve for the unknowns.

• Write position equations for both objects:

• $x_A = x_{0A} + v_{0A} t_A + \frac{1}{2} a_A t_A^2$

• $x_B = x_{0B} + v_{0B} t_B + \frac{1}{2} a_B t_B^2$

• Set $x_A = x_B$ and solve for $t$ or other variables as needed.

Summary Table: Graphical Relationships in Kinematics

Graph	Y-axis	X-axis	Slope	Area
Position-Time (x-t)	Position (x)	Time (t)	Velocity (v)	—
Velocity-Time (v-t)	Velocity (v)	Time (t)	Acceleration (a)	Displacement (Δx)
Acceleration-Time (a-t)	Acceleration (a)	Time (t)	—	Change in velocity (Δv)

Key Problem-Solving Steps for Kinematics

1. Draw a diagram and list all known and unknown variables.

2. Identify which kinematics equation(s) to use based on the variables involved.

3. Solve algebraically for the target variable.

4. Check units and physical reasonableness of your answer.