Motion

Graphs & Motion of a Particle

Fundamentals of Motion and Motion Diagrams

A description of motion encompasses both place (where the object is positioned) and time.

A Motion Diagram is a series of images showing the positions of a moving object across successive intervals of time. It visually displays the motion of an object over time.

The Particle Model and Position Vectors

The Particle Model is a modified motion diagram in which a series of single points (or dots) replaces the object in motion.

• Position: Defined as the separation between an object and the origin.

• Origin and Vectors: In a diagram where the origin is set at the left, dots represent the object (such as a runner). An arrow represents the position of the moving object, pointing directly from the origin to the object at a particular time.

Particle Model Applications

The particle model is used to construct motion diagrams for various movement scenarios:

• A runner starting off at a sprint and then slowing down.

• A runner that starts from rest, speeds up, and then slows to a stop.

• Three runners in a race at a specific moment: the first runner crosses the finish line, the 2nd runner is three-fourths of the way to the finish line, and the 3rd runner is still close to the starting point.

Position-Time Graphs

A position-time graph (position vs. time graph) is created by plotting the known position of an object or runner at various points in time.

• Line of Best Fit: After plotting the points, a line of best fit is drawn. The line of best fit reveals the positions of the runners at each data point.

• Interpretative Functions: A position-time graph is used to interpret an object's position or displacement. It allows one to determine both where and when an object or person is located.

Vectors, Displacement, & the Motion of a Particle

Fundamentals of Vectors and Scalars

Vectors represent quantities that have magnitude (size) and direction and can be represented by arrows. They are measurements that possess both magnitude and direction.

Scalars represent quantities such as distance, time, or temperature without direction.

Kinematic Quantities and Mathematical Formulations

Displacement represents a change in position:

$\Delta d = d_f - d_i$

Distance represents how far the object is from the origin. Distance is a scalar quality.

Time interval is the difference between $2$ times:

$\Delta t = t_f - t_i$

Average velocity is determined as the change in position, divided by time.

Average speed is the ratio of the total distance traveled divided by the time interval.

Acceleration is the change in velocity during measurable time intervals divided by the time interval it takes to make that change. Average acceleration is measured in $m/s^2$.

Graphical Analysis and Position-Time Relationships

Position-time graphs contain the velocity, displacement, and position of an object and every moment of time.

Both displacement and time interval are needed to tell how fast an object is moving.

When comparing the lines representing Runner 1 and Runner 2 in a position-time graph:

• The slope of Runner 1 is steeper than the slope of Runner 2.

• A steep slope is indicative of a greater change in displacement during the time interval.

• Runner 1 is moving faster than Runner 2.

Questions & Discussion

Question: The graph describes the motion of Runner 3 through a park. What is his average velocity? What is his average speed?

Response/Method: Find the average velocity of the runner using two points on the line.

Uniform & Accelerated Motion with Scalars & Vectors

Fundamentals of Velocity and Acceleration

The movement of objects is frequently non-uniform rather than occurring at a constant velocity. Modes of transportation such as planes, subways, and cars typically commence motion with rapid speed before coming to either a speedy or slow halt. The physical motion of an object is directly governed by both the direction and magnitude of its velocity and acceleration.

Velocity and acceleration are defined by rates of change over time:

• Velocity represents the time rate of change of position.

• Acceleration represents the time rate of change of velocity.

The directional interaction between velocity and acceleration dictates whether an object speeds up or slows down:

• When velocity and acceleration act in the same direction, the object increases speed.

• When velocity and acceleration act in opposite directions, the object slows down.

Acceleration on Velocity-Time Graphs

On a velocity-time graph, acceleration corresponds directly to the slope of the plotted curve.

• Average Acceleration: Defined as the change in velocity divided by the time elapsed during that change.

• Instantaneous Acceleration: Defined as the change in velocity at an exact instant in time. Instantaneous acceleration is determined geometrically by drawing a tangent line to the velocity-time curve at the specific point of interest and calculating its slope.

Sprinter Motion Analysis and Calculations

A velocity-time profile of a sprinter illustrates non-constant accelerated motion. The sprinter begins at an initial velocity of $0\,\text{m/s}$, undergoes a rapid increase in velocity over the first few seconds, and subsequently levels off to maintain an approximately constant speed after reaching $10\,\text{m/s}$.

Evaluating the instantaneous acceleration at selected time intervals requires finding the slope ($\frac{\text{rise}}{\text{run}}$) of the tangent line at each point:

Instantaneous Acceleration at $t = 1.0\,\text{s}$:

$a = \frac{\text{rise}}{\text{run}}$

$a = \frac{11.0\,\text{m/s} - 2.8\,\text{m/s}}{2.40\,\text{s} - 0.00\,\text{s}}$

$a = 3.4\,\text{m/s}^2$

Instantaneous Acceleration at $t = 5.0\,\text{s}$:

$a = \frac{\text{rise}}{\text{run}}$

$a = \frac{10.3\,\text{m/s} - 10.0\,\text{m/s}}{10.0\,\text{s} - 0.00\,\text{s}}$

$a = 0.030\,\text{m/s}^2$

Analysis of Results:

• The acceleration is not constant, as it decreases from $3.4\,\text{m/s}^2$ at $t = 1.0\,\text{s}$ to $0.03\,\text{m/s}^2$ at $t = 5.0\,\text{s}$.

• The acceleration acts in a positive direction, demonstrated by the positive numerical values calculated.

Projectile Motion

Fundamentals of Projectile Motion

A projectile is an object that is shot through the air under the force of gravity. The object is given an initial thrust, ignores air resistance, and subsequently moves downward under the action of the force of gravity.

Examples of projectiles include:

• A football

• A bullet

• A drop of water

Trajectory and Motion Dynamics

The trajectory is the path of a projectile through space. A trajectory can be calculated if the force of the initial thrust is known.

Key physical properties governing projectile flight include:

• Parabola: The curved flight path traced by a projectile is called a parabola.

• Independence of Motion: The vertical and horizontal motions of a projectile are independent of each other.

• Range Factors: The range of a projectile depends on the acceleration resulting from gravity and both components of the initial velocity.

Solving Projectile Problems

Projectile motion problems are systematically solved by separating the vertical and horizontal components of motion:

1. Vertical Motion Analysis: Use the vertical motion component to relate the initial vertical velocity, height, and total time in the air.

2. Horizontal Motion Analysis: Once the time in the air is established, use it to calculate the range and the horizontal distance traveled.

Uniform Circular & Rotary Motion

Fundamentals of Uniform Circular Motion

Uniform circular motion is defined as the movement of an object or particle trajectory at a constant speed around a circle with a fixed radius.

The position vector defines the position of an object in uniform circular motion relative to the center of the circle. As the object moves along its circular path, the direction of the position vector continuously changes, but the length of the position vector remains constant.

Displacement and velocity in uniform circular motion:

• Object displacement or change in position is represented by $\Delta r$.

• The average velocity of an object during a time interval is equal to the displacement vector divided by the time interval:

$\text{Object's velocity} = \frac{\Delta r}{\Delta t}$

Centripetal Acceleration Principles and Equations

An object moving in a circle at a constant speed undergoes constant acceleration directed toward the center of the circle, known as centripetal acceleration ($a_c$).

Centripetal acceleration characteristics:

• Direction: Always points directly toward the center of the circle.

• Dependence: Varies directly with the square of the object's speed and inversely with the radius of the circular path.

Formulas for centripetal acceleration:

• Expressed in terms of linear speed ($v$) and radius ($r$):

$a_c = \frac{v^2}{r}$

• Expressed in terms of radius ($r$) and the period of revolution ($T$):

$a_c = \frac{4\pi^2 r}{T^2}$

Circular Motion Problem Solving: String Tension

Problem Statement

A $26\,\text{g}$ stopper is attached to a $1.86\,\text{m}$ string. The stopper is swung in a horizontal circle, completing one revolution in $1.38\,\text{s}$. Find the tension force exerted by the string on the stopper.

Procedure

1. Analyze and sketch the problem scenario, including the circle's radius and the direction of circular motion.

2. Identify known and unknown variables.

3. Compute the centripetal acceleration and corresponding tension force.

Given Data

• Mass: $\text{mass} = 26\,\text{g}$

• Radius: $r = 1.86\,\text{m}$

• Period: $T = 1.38\,\text{s}$

Unknown Quantity

• Tension force: $F_T$

Calculations

Calculate centripetal acceleration using the period and radius:

$a_c = \frac{4\pi^2 r}{T^2}$

Motion of Fluids

Fundamental Fluid Properties and Concepts

Fluids are materials that flow and have no definite shape of their own. The particles in a fluid are constantly moving and exert pressure as they collide with all surfaces in contact with the fluid.

Pressure is the force on a surface divided by the area of the surface. It is a scalar quantity, and the SI unit of pressure is the Pascal, $\text{Pa}$.

$P = \frac{F}{A}$

Viscosity is the internal friction that slows down the movement of a fluid. It represents the resistance of a fluid to flow; for example, engine oil and molasses are very viscous.

Surface tension is the tendency of the surface of a liquid to contract to the smallest possible area. The higher the surface tension of the liquid, the harder it is for the surface of the liquid to break. An example of this phenomenon is a water strider walking on water.

Capillary action refers to the movement of water caused by its adhesive forces up a tube. Examples include the way water moves up through the soil, up the xylem of a plant, and up narrow tubes.

Phase Changes in Fluids

Evaporation occurs when the most energetic particles of a liquid have enough energy to convert to the gaseous phase. This is the process through which a liquid becomes a gas.

Condensation occurs when the particles within a gas with the least amount of energy bind to each other and convert to liquid form.

Fluid Dynamics and Governing Principles

Pascal's principle states that an applied pressure is transmitted undiminished through a fluid. In an enclosed fluid, pressure changes at any point are transmitted undiminished at every point and act in all directions in the fluid.

Bernoulli's principle states that as the velocity of a fluid increases, the pressure exerted by the fluid decreases.

Archimedes' Principle and Buoyant Force

Archimedes' principle states that the buoyant force on an object immersed in a fluid is equal to the weight of the fluid displaced by that object. An upward force exerted on an object immersed in a fluid is called a buoyant force.

The buoyant force is equal to the density of the fluid in which the object is immersed, multiplied by the object's volume and the acceleration due to gravity.

$\text{Buoyant Force} = F_{\text{buoyant}} = P_{\text{water}} V g$

Applications of Fluids and Thermal Expansion

Fluids exert forces that contribute to everyday life in activities and technologies such as swimming, diving, floating, flying planes, and balloons.

Thermal expansion provides principles that assist in the engineering and structural design of buildings, roads, bridges, and machines.

Frames of Reference & Relativity

Definition and Classification of Frames of Reference

A frame of reference is a set of coordinates that help to determine the motion and positions of a moving object within that frame.

Reference frames are divided into inertial and noninertial types:

• Non-accelerating (inertial) reference frames: Apply to Newton’s laws of motion. In these frames, an object moves at a constant speed in a straight line until it experiences an unbalanced force.

• Accelerated frame (Non-inertial): Newton’s laws are invalid with respect to a rotating frame of reference, which is an accelerated frame.

Observer Perspective and Relative Velocity

Observers situated in different frames of reference may view the same phenomenon differently:

• A person traveling on a plane may observe an object flying through the air, yet the velocity of the object appears to be $0$.

• To an observer viewing the same event through a telescope, the object appears to be moving at about $200\,\text{m/s}$.

Each viewer has a different frame of reference. In this situation, the relative velocity of the two frames is $200\,\text{m/s}$. The frames are said to be equal because Newton’s laws can analyze motion in either frame.

Theories of Relativity and Frame Dependence

Relativity is divided into two distinct theories:

• Special Relativity: Concerns motion at constant velocity.

• General relativity: Concerns accelerated motion and motion in a gravitational field.

The importance of relativity is that physical values and observations are only true in one frame of reference. Velocity and time are primary examples of properties that can change based on the observer's frame of reference.