Motion along a Straight Line: Physics I Study Notes

Learning Outcomes

• Define position, displacement, velocity, speed, and acceleration.
• Distinguish between vector and scalar kinematic quantities.
• Use the definitions of average velocity and acceleration to perform simple computations.
• State the definition of free fall and identify common characteristics of free-falling objects.
• Apply the kinematic equations for 1D motion of particles/objects.
• Interpret displacement-time, velocity-time, and acceleration-time graphs.
• Apply a simple mathematical model to solve one-dimensional and two-dimensional motion problems (CLO 2).

Study of Motion: Kinematics vs. Dynamics

• Kinematics: Often referred to as the "geometry of motion," kinematics deals with pure motion without reference to the masses or forces involved in it. It is concerned with describing how objects move.
• Dynamics: Dynamics is concerned with the study of forces and torques and their effect on motion. It is concerned with explaining why motion occurs.
• Both Kinematics and Dynamics are branches of Mechanics.

Displacement and Average Velocity

• Relative Position: When a particle moves along a straight line (e.g., the x-axis), its position is described by a single coordinate $x$.
• Displacement: The change in position of a particle. If at time $t_1$ a point is at position $x_1$ and at time $t_2$ it is at position $x_2$, the displacement is the vector from $x_1$ to $x_2$. The x-component of the displacement is given by:   $\Delta x = x_2 - x_1$
• Average Velocity ($v_{av,x}$): A vector defined in the direction from $x_1$ to $x_2$. The x-component is calculated as:   $v_{av,x} = \frac{x_2 - x_1}{t_2 - t_1} = \frac{\Delta x}{\Delta t}$
• Units: The SI unit for velocity is meters per second ($m/s$).
• Displacement-Time Graph:   - Position $x$ is a function of time $t$.   - The average velocity between two positions is represented by the slope of a line (secant line) connecting the two corresponding points on a graph of position as a function of time.

Example: Travel from Abu Dhabi to Dubai

• Scenario: A vehicle leaves Abu Dhabi at 09:45 AM and arrives in Dubai at 11:20 AM. The distance is $135\,km$.
• Time Interval Calculation:   - $09:45\,AM$ to $11:20\,AM = 1\,hour\,35\,minutes = 95\,minutes$.   - $95\,min \times 60\,s/min = 5700\,s$.
• Distance Conversion:   - $135\,km = 135,000\,m$.
• Average Velocity Calculation:   - $v_{av,x} = \frac{135,000\,m}{5700\,s} = 23.7\,m/s$.

Example: Ant Crawling Exercise

• Scenario: An ant crawls along a wire (x-axis) through points A, B, C, and D. $O$ is the origin. Measurements: $AB = 50\,cm$, $BC = 30\,cm$, and $AO = 5\,cm$. Point $D$ overlaps point $A$.
• Positions:   - $x_A = -5\,cm$   - $x_B = x_A + 50\,cm = 45\,cm$   - $x_C = x_B - 30\,cm = 15\,cm$   - $x_D = -5\,cm$
• Interval Analysis:   - A to B: Displacement = $+50\,cm$, Distance = $\text{50 cm}$.   - B to C: Displacement = $-30\,cm$, Distance = $\text{30 cm}$.   - C to D: Displacement = $x_D - x_C = -5 - 15 = -20\,cm$, Distance = $\text{20 cm}$.   - A to D: Displacement = $x_D - x_A = -5 - (-5) = 0\,cm$, Distance = $50 + 30 + 20 = 100\,cm$.

Instantaneous Velocity and Speed

• Definition: The instantaneous velocity at any point on a graph of coordinate $x$ vs. time $t$ is equal to the slope of the tangent line to the curve at that specific point.
• Instantaneous Speed: The magnitude of the instantaneous velocity at a specific moment.
• Technology Example: Instantaneous speed cameras measure speed at a specific point, whereas average speed cameras calculate speed over a distance between two points.

Average and Instantaneous Acceleration

• Definition: Acceleration is a quantitative description of the rate of change of velocity with time.
• Average Acceleration ($a_{av,x}$): If at time $t_1$ a particle has velocity $v_{1x}$ and at time $t_2$ it has velocity $v_{2x}$, the average acceleration is:   $a_{av,x} = \frac{v_{2x} - v_{1x}}{t_2 - t_1} = \frac{\Delta v_x}{\Delta t}$
• Instantaneous Acceleration ($a_x$): The limit approached by the average acceleration as the time interval approaches zero ($\Delta t \to 0$). It is the slope of the tangent line to the curve on a velocity-time ($v-t$) graph.
• Positive and Negative Acceleration:   - If velocity and acceleration have the same sign, the object is speeding up.   - If velocity and acceleration have opposite signs, the object is slowing down.

Graphical Representation of Motion

• Types of Graphs:   1. Displacement-time ($x-t$)   2. Velocity-time ($v-t$)   3. Acceleration-time ($a-t$)
• Slope Relationships:   - The slope of an $x-t$ graph equals velocity.   - The slope of a $v-t$ graph equals acceleration.
• Area Relationships:   - The area under a $v-t$ graph equals the displacement (or distance) travelled.

Motion with Constant Acceleration

• If acceleration is constant, velocity changes at the same rate throughout the motion.
• Kinematic Equations:   1. $v_x = v_{0x} + a_x t$   2. $x = x_0 + v_{0x} t + \frac{1}{2} a_x t^2$   3. $v_x^2 = v_{0x}^2 + 2 a_x (x - x_0)$   4. $x - x_0 = \frac{1}{2} (v_{0x} + v_x) t$
• Where:   - $v_{0x}$ = Initial velocity   - $v_x$ = Final velocity   - $a_x$ = Constant acceleration   - $x_0$ = Initial position   - $x$ = Final position   - $t$ = Time elapsed

Example: Passing Speed

• Scenario: A car traveling at $15\,m/s$ accelerates at $2.0\,m/s^2$ for $5.0\,s$.
• Calculations:   - Final Velocity: $v_x = 15 + (2.0 \times 5.0) = 25\,m/s$.   - Distance Travelled: $x - x_0 = (15 \times 5.0) + \frac{1}{2} (2.0) (5.0^2) = 75 + 25 = 100\,m$.

Example: Traffic Signal Braking

• Scenario: A car at $20\,m/s$ brakes to a stop in $5.0\,s$.
• Calculations:   - Acceleration: $a_x = \frac{0 - 20}{5.0} = -4.0\,m/s^2$.   - Distance to line: $x - x_0 = \frac{1}{2} (20 + 0) \times 5.0 = 50\,m$.

Example: Ball Rebounding off Wall

• Scenario: A ball hits a wall horizontally at $4.8\,m/s$ and rebounds at $3.6\,m/s$. Contact time is $280\,ms$.
• Calculations:   - Change in velocity: $\Delta v = v_{final} - v_{initial} = (-3.6) - 4.8 = -8.4\,m/s$.   - Acceleration: $a = \frac{-8.4\,m/s}{0.280\,s} = -30\,m/s^2$.

Free Fall Motion

• Core Definition: Free fall is motion with constant acceleration under the influence of Earth's gravity, where air resistance is neglected.
• Galileo’s Discovery: In the absence of air resistance, all objects fall with the same constant acceleration regardless of their weight/mass.
• Acceleration due to Gravity ($g$):   - Magnitude near Earth's surface: $g \approx 9.8\,m/s^2$.   - Magnitude on the Moon: $g \approx 1.6\,m/s^2$.
• Conventions for Equation Application:   - Acceleration in the y-direction ($a_y$) is usually taken as $-g$ (where upward is positive).   - At the highest point of a trajectory, the vertical velocity component is $0$.

Example: Tennis Serving Machine

• Scenario: Ball shot vertically UP at $30\,m/s$.
• Graphs Analysis:   - Acceleration: Continuous horizontal line at $-9.8\,m/s^2$.   - Velocity: Downward sloping straight line starting at $+30$, crossing zero at the peak, and reaching $-30$ upon return.   - Position: Parabola opening downwards (inverted U-shape).

Example: Ball Thrown from Building Roof

• Scenario: Ball thrown upward from a roof with $v_{0y} = 15.0\,m/s$. Position $y_0 = 0$.
• Calculations for 1.00 s:   - Velocity: $v_y = 15.0 + (-9.8 \times 1.0) = 5.2\,m/s$.   - Position: $y = (15.0 \times 1.0) + \frac{1}{2} (-9.8) (1.0^2) = 10.1\,m$.
• Calculations for 4.00 s:   - Velocity: $v_y = 15.0 + (-9.8 \times 4.0) = -24.2\,m/s$.   - Position: $y = (15.0 \times 4.0) + \frac{1}{2} (-9.8) (4.0^2) = 60 - 78.4 = -18.4\,m$.
• Calculations for Maximum Height:   - Velocity at max height = $0$.   - Time to reach top: $t = \frac{0 - 15.0}{-9.8} = 1.53\,s$.   - Max Height: $y = (15.0 \times 1.53) + \frac{1}{2} (-9.8) (1.53^2) = 11.5\,m$.

Summary Key Points

• Displacement segments $\Delta y$ and $\Delta x$ are interchangeable based on the axis of motion.
• In free fall problems, always assume $a_y = -g = -9.8\,m/s^2$.
• Launch points are typically defined as $t=0$, $x=0$, and $y=0$.
• Reading assignment: Section 2.7 regarding Relative Velocity along a straight line (excluding the Theory of Relativity section).