Kinematics: One-Dimensional and Two-Dimensional Motion

One-Dimensional Motion

Introduction to Kinematics

Kinematics is the branch of physics that describes the motion of objects without considering the causes of motion. In one-dimensional motion, objects move along a straight line, and their position, velocity, and acceleration can be described using simple equations.

• Position (x): The location of an object along a straight line.

• Displacement (Δx): The change in position of an object.

• Velocity (v): The rate of change of position with respect to time.

• Acceleration (a): The rate of change of velocity with respect to time.

Acceleration

Acceleration occurs when the velocity of an object changes over time. It is a vector quantity, meaning it has both magnitude and direction.

• Definition: Acceleration is the rate of change of velocity.

• Formula: $a = \frac{dv}{dt}$

• Example: When you press the accelerator in a car, you increase its velocity, causing acceleration.

Constant Acceleration Equations

When acceleration is constant, the following kinematic equations can be used to solve problems:

• $v = v_0 + at$

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

• $v^2 = v_0^2 + 2a(x - x_0)$

Where:

• $v_0$ = initial velocity

• $v$ = final velocity

• $a$ = acceleration

• $t$ = time

• $x_0$ = initial position

• $x$ = final position

Stopping a Car: Constant Deceleration

When a car traveling at an initial speed $v_0$ comes to a stop with constant deceleration $a$, you can calculate:

• Time to stop: $t = \frac{v_0}{a}$

• Distance traveled before stopping: $x = \frac{v_0^2}{2a}$

Example: If a car moving at 20 m/s decelerates at 5 m/s2, it will stop in 4 seconds and travel 40 meters before stopping.

Chasing a Speeder: Relative Motion

If a police officer starts from rest and accelerates at a constant rate $a$ to catch a speeder moving at constant velocity $V$, you can determine:

• Time to catch up: Set the positions equal and solve for $t$.

• Distance traveled: Use $x = \frac{1}{2} a t^2$ for the officer and $x = V t$ for the speeder.

• Final speed: $v = a t$ for the officer at the moment of catching up.

Derivation: Kinematic Equation

For constant acceleration, the following equation relates displacement, acceleration, and velocities:

• $2a(\Delta x) = v_f^2 - v_0^2$

This equation is derived by eliminating time from the basic kinematic equations.

Free Fall

Free fall is a classic example of one-dimensional motion under constant acceleration due to gravity.

• Acceleration due to gravity: $a = g = 9.8\,\text{m/s}^2$ downward

• Objects in free fall: All objects accelerate towards Earth at the same rate, regardless of mass (ignoring air resistance).

Throwing a Ball Upward

When a ball is thrown upward with initial velocity $v_0$, several quantities can be calculated:

• Time to reach highest point: $t = \frac{v_0}{g}$

• Maximum height: $h = \frac{v_0^2}{2g}$

• Total time to return to hand: $t_{total} = 2 \frac{v_0}{g}$

• Velocity when it returns: $-v_0$ (same magnitude, opposite direction)

• Time to reach height $h$: Solve $h = v_0 t - \frac{1}{2} g t^2$ for $t$

Key Concept: Acceleration due to gravity is constant throughout the motion, regardless of the ball's position.

Two-Dimensional Motion

Projectile Motion

Projectile motion involves two-dimensional kinematics, where the horizontal and vertical motions are independent except for sharing the same time of flight.

• Horizontal motion: Constant velocity ($a_x = 0$)

• Vertical motion: Constant acceleration ($a_y = -g$)

• Equations:

  • $x = x_0 + v_{0x} t$

  • $y = y_0 + v_{0y} t - \frac{1}{2} g t^2$

  • $v_x = v_{0x}$

  • $v_y = v_{0y} - g t$

• Problem-solving tip: Break the motion into x and y components and solve separately.

Independence of Motion

In projectile motion, the horizontal and vertical motions do not affect each other. For example, if two marbles are released simultaneously—one horizontally and one vertically—they hit the ground at the same time (assuming equal heights and no air resistance).

Monkey and Dart Problem

If a dart is fired at a monkey that lets go of a branch at the same instant, the dart should be aimed directly at the monkey. Both the dart and the monkey experience the same vertical acceleration due to gravity, so their paths intersect.

• Key Point: Gravity affects both objects equally, so aim directly at the target.

Projectile Range and Angle

The horizontal distance (range) traveled by a projectile depends on the launch angle and initial speed. The range is maximized at a launch angle of 45 degrees (for level ground).

• Range formula: $R = \frac{v_0^2 \sin(2\theta)}{g}$

• Maximum range: At $\theta = 45^\circ$

Relative Motion

Relative motion describes how the velocity of an object appears from different reference frames. The observed velocity depends on the motion of both the observer and the object.

• General formula: $\vec{v}_{A/C} = \vec{v}_{A/B} + \vec{v}_{B/C}$

• Example: If a dog runs at 8 m/s on a sidewalk moving at 2 m/s, the dog's speed relative to the ground is $8 + 2 = 10$ m/s if moving in the same direction, or $8 - 2 = 6$ m/s if in opposite directions.

Crossing a River

When swimming across a river with a current, the swimmer's velocity relative to the ground is the vector sum of their velocity relative to the water and the water's velocity.

• Time to cross: Depends only on the component of swimmer's velocity perpendicular to the current.

• Angle to aim: To reach a point directly across, the swimmer must aim upstream at an angle $\theta$ such that $v_{swimmer} \sin(\theta) = v_{current}$.

Table: Relative Velocities in Different Scenarios

Scenario	Velocity Relative to Ground	Velocity Relative to Observer
Dog runs on moving sidewalk (same direction)	10 m/s	8 m/s
Dog runs on moving sidewalk (opposite direction)	6 m/s	8 m/s
Swimmer crosses river perpendicular to current	Resultant velocity	Swimmer's speed
Swimmer aims upstream/downstream	Depends on angle	Swimmer's speed

Additional info: Some context and equations have been expanded for clarity and completeness, including the derivation of kinematic equations and the explanation of relative motion scenarios.