Chapter 2: Motion in One Dimension – Structured Study Notes

Motion in One Dimension

Introduction to Linear Motion

Motion in one dimension refers to the movement of objects along a straight line, either horizontally or vertically. This chapter focuses on describing, analyzing, and solving problems related to linear motion using graphical, pictorial, and mathematical representations.

• Key Kinematic Variables: Position, velocity, and acceleration.

• Types of Motion: Uniform motion (constant velocity) and motion with constant acceleration (including free fall).

• Problem-Solving Skills: Strategies for approaching and solving physics problems.

Describing Motion

Representing Position

Position is the location of an object relative to a chosen origin. In one-dimensional motion, position is measured along an axis (x for horizontal, y for vertical).

• Positive and Negative Positions: Positions to the right or above the origin are positive; positions to the left or below are negative.

• Coordinate Axes: Used to analyze motion direction and magnitude.

Position vs. Time Graphs

Graphs of position versus time provide a visual representation of how an object's position changes over time. The slope of the graph indicates the speed and direction of motion.

• Steeper Slope: Indicates faster speed.

• Positive Slope: Motion to the right/up.

• Negative Slope: Motion to the left/down.

Interpreting Position Graphs

Position graphs can be used to describe an object's motion in words, such as whether it is moving, stopped, or changing direction.

• Constant Position: Object is at rest.

• Changing Position: Object is moving.

• Direction Change: Slope changes sign.

Velocity and Its Representation

From Position to Velocity

Velocity is the rate of change of position with respect to time. On a position-versus-time graph, velocity is represented by the slope.

• Formula: $v = \frac{\Delta x}{\Delta t}$

• Steeper Slope: Higher velocity.

• Sign of Slope: Indicates direction.

Velocity vs. Time Graphs

Velocity-versus-time graphs provide another way to represent motion. The area under the velocity graph gives the displacement.

• Constant Velocity: Horizontal line.

• Changing Velocity: Sloped line.

• Displacement: Area under the curve.

Example: Finding Velocity from Position Graph

To find velocity from a position graph, measure the slope for each segment.

• Example Calculation: $v = \frac{\Delta x}{\Delta t}$ for each interval.

• Interpretation: Negative velocity means moving backward; zero velocity means at rest; positive velocity means moving forward.

Uniform Motion

Definition and Characteristics

Uniform motion occurs when an object moves in a straight line with constant velocity. The position-versus-time graph is a straight line.

• Position Equation: $x_f = x_i + v \Delta t$

• Displacement: $\Delta x = v \Delta t$

Example: Ratio Reasoning in Uniform Motion

When an object moves at constant speed, the distance traveled is proportional to the time elapsed.

• Example: If a train travels 12 km in 10 min, it will travel 60 km in 50 min.

Instantaneous Velocity

Definition and Calculation

Instantaneous velocity is the velocity of an object at a specific instant. It is found by calculating the slope of the tangent to the position-versus-time curve at that point.

• Graphical Method: Draw a tangent line and calculate its slope.

• Formula: $v = \frac{dx}{dt}$ (for calculus-based courses; for algebra-based, use small intervals).

Acceleration

Definition and Representation

Acceleration is the rate of change of velocity with respect to time. It is represented by the slope of the velocity-versus-time graph.

• Formula: $a = \frac{\Delta v}{\Delta t}$

• Positive Acceleration: Speeding up in the positive direction.

• Negative Acceleration: Slowing down or speeding up in the negative direction.

Motion with Constant Acceleration

Kinematic Equations

For motion with constant acceleration, several equations relate position, velocity, acceleration, and time.

• Velocity Equation: $v_f = v_i + a \Delta t$

• Position Equation: $x_f = x_i + v_i \Delta t + \frac{1}{2} a (\Delta t)^2$

• Velocity-Position Relation: $v_f^2 = v_i^2 + 2a \Delta x$

Free Fall

Definition and Properties

Free fall describes the motion of objects under the influence of gravity alone. All objects in free fall experience the same acceleration, regardless of mass.

• Free-Fall Acceleration: $a = g = 9.80\ \text{m/s}^2$ (downward)

• Motion Equations: Use kinematic equations with $a = -g$ for upward motion.

Problem-Solving Approach

Four-Step Method

Effective problem-solving in physics involves four steps: strategize, prepare, solve, and assess.

• Strategize: Identify the type of problem and approach.

• Prepare: Draw diagrams, set up coordinate systems, and gather information.

• Solve: Apply equations and perform calculations.

• Assess: Check if the solution makes sense physically and mathematically.

Summary Table: Kinematic Variables and Equations

Variable	Definition	Equation
Position (x)	Location relative to origin	—
Velocity (v)	Rate of change of position	$v = \frac{\Delta x}{\Delta t}$
Acceleration (a)	Rate of change of velocity	$a = \frac{\Delta v}{\Delta t}$
Uniform Motion	Constant velocity	$x_f = x_i + v \Delta t$
Constant Acceleration	Velocity changes steadily	$v_f = v_i + a \Delta t$ $x_f = x_i + v_i \Delta t + \frac{1}{2} a (\Delta t)^2$ $v_f^2 = v_i^2 + 2a \Delta x$
Free Fall	Motion under gravity	$a = -g$

Graphical Representations

Summary of Graphs

Understanding motion requires interpreting position, velocity, and acceleration graphs. Key features include slope, area under curves, and direction of change.

Applications and Examples

Example: Rocket Launch

A rocket launched with constant acceleration can be analyzed using kinematic equations to find its velocity and position after a given time.

• Given: $a = 18\ \text{m/s}^2$, $t = 150\ \text{s}$

• Find: $v_f = 0 + 18 \times 150 = 2700\ \text{m/s}$

• Find: $y_f = 0 + 0 \times 150 + \frac{1}{2} \times 18 \times (150)^2 = 2.0 \times 10^5\ \text{m}$

Example: Minimum Runway Length

To calculate the minimum runway length for a plane, use the constant acceleration equations with the given acceleration and required takeoff speed.

Example: Free Fall of a Rock

When a rock is dropped from a height, its fall time and impact velocity can be found using the free fall equations.

• Given: $y_i = 100\ \text{m}$, $v_i = 0$, $a = -9.8\ \text{m/s}^2$

• Find: $t = \sqrt{\frac{2y}{g}} = 4.52\ \text{s}$

• Find: $v_f = -9.8 \times 4.52 = -44.3\ \text{m/s}$

Example: Springbok Jump

The height and speed of a springbok's jump can be calculated using the kinematic equations for constant acceleration and free fall.

• Lift-off Speed: $v_f = \sqrt{2a\Delta y} = 7.0\ \text{m/s}$

• Jump Height: $y_f = \frac{v_i^2}{2g} = 2.5\ \text{m}$

Conclusion

Motion in one dimension is foundational for understanding physics. Mastery of position, velocity, and acceleration, along with their graphical and mathematical representations, enables students to analyze and solve a wide range of physical problems.