Motion in a Straight Line Flashcards

INTRODUCTION TO RECTILINEAR MOTION

• Universal Nature of Motion: Motion is a fundamental property of everything in the universe. Examples include:

  • Human activities: Walking, running, and riding a bicycle.

  • Biological processes: Air moving in and out of lungs and blood flowing in arteries and veins.

  • Natural phenomena: Leaves falling from trees and water flowing down a dam.

  • Transportation: Automobiles and planes moving people.

  • Astronomical motion: The Earth rotates once every $24\,\text{hours}$ and revolves around the Sun once a year. The Sun moves within the Milky Way, which moves within its local group of galaxies.

• Definition of Motion: Motion is defined as the change in position of an object with time.

• Rectilinear Motion: This refers specifically to the study of the motion of objects along a straight line.

• Point Object Approximation: An object in motion is treated as a point object if its size is much smaller than the distance it moves over a reasonable duration of time. This approximation allows for simplified descriptions of motion in real-life situations without significant error.

• Kinematics: The branch of physics that studies ways to describe motion without considering the causes of that motion. The causes of motion (dynamics) are discussed in later chapters.

INSTANTANEOUS VELOCITY AND SPEED

• Average vs. Instantaneous Velocity: While average velocity describes the overall motion over a time interval, it does not provide information about how fast an internal object moves at specific moments. Instantaneous velocity (or simply velocity $v$) provides the speed and direction at a specific instant $t$.

• Mathematical Definition: Instantaneous velocity is the limit of the average velocity as the time interval $\Delta t$ becomes infinitesimally small:     $v = \lim_{\Delta t \rightarrow 0} \frac{\Delta x}{\Delta t} = \frac{dx}{dt}$

  • In calculus, this is known as the differential coefficient of $x$ with respect to $t$, representing the rate of change of position with respect to time at that instant.

• Graphical Determination: Velocity at a specific time $t$ is the slope of the tangent to the position-time ($x-t$) graph at that point.

• Numerical Example (Table 2.1): For a car whose position is defined by $x = 0.08 t^3$, the velocity at $t = 4.0\,s$ is determined by decreasing $\Delta t$ from $2.0\,s$ to $0.01\,s$. The ratio $\frac{\Delta x}{\Delta t}$ approaches the limiting value of $3.84\,m\,s^{-1}$, which is the instantaneous velocity at that point.

• Instantaneous Speed: This is the magnitude of the instantaneous velocity. Unlike average values, where average speed can be greater than the magnitude of average velocity, instantaneous speed is always equal to the magnitude of instantaneous velocity at that instant.

• Example 2.1: Given position $x = a + bt^2$ with $a = 8.5\,m$ and $b = 2.5\,m\,s^{-2}$.

  • Velocity $v = \frac{dx}{dt} = 2bt = 5.0t\,m\,s^{-1}$.

  • At $t = 0\,s$, $v = 0\,m\,s^{-1}$.

  • At $t = 2.0\,s$, $v = 10\,m\,s^{-1}$.

  • Average velocity between $2.0\,s$ and $4.0\,s$: $\frac{x(4.0) - x(2.0)}{4.0 - 2.0} = 15\,m\,s^{-1}$.

ACCELERATION

• Definition: Acceleration ($a$) is the rate of change of velocity with time. Galileo concluded through studies of free fall and inclined planes that velocity change with respect to time is constant for free fall, whereas change with respect to distance is not.

• Average Acceleration: The change in velocity divided by the time interval:     $\bar{a} = \frac{v_2 - v_1}{t_2 - t_1} = \frac{\Delta v}{\Delta t}$

  • The SI unit is $m\,s^{-2}$.

• Instantaneous Acceleration: The limit of average acceleration as $\Delta t \rightarrow 0$:     $a = \lim_{\Delta t \rightarrow 0} \frac{\Delta v}{\Delta t} = \frac{dv}{dt}$

  • Graphically, it is the slope of the tangent to the velocity-time ($v-t$) curve at a specific instant.

• Characteristics of Acceleration:

  • It can be positive, negative, or zero.

  • In an $x-t$ graph: it curves upward for positive acceleration, downward for negative acceleration, and is a straight line for zero acceleration.

  • A change in velocity may involve change in speed (magnitude), change in direction, or both.

• Area Under the Curve: In a velocity-time graph, the area under the curve represents the displacement of the object over a specific time interval.

KINEMATIC EQUATIONS FOR UNIFORMLY ACCELERATED MOTION

For motion with constant acceleration ($a$), initial velocity ($v_0$), final velocity ($v$), time ($t$), and displacement ($x$):

1. Velocity-Time Relation:     $v = v_0 + at$

2. Displacement-Time Relation:     $x = v_0 t + \frac{1}{2}at^2$

   • If the initial position is $x_0$ instead of $0$, then: $x = x_0 + v_0 t + \frac{1}{2}at^2$

3. Velocity-Displacement Relation:     $v^2 = v_0^2 + 2ax$

   • Or: $v^2 = v_0^2 + 2a(x - x_0)$

4. Arithmetic Average Velocity: For constant acceleration, the displacement can also be calculated as:     $x = \frac{v + v_0}{2} t$

• Calculus Derivation:

  • $a = \frac{dv}{dt} \rightarrow \int_{v_0}^{v} dv = \int_{0}^{t} a \, dt \rightarrow v - v_0 = at$

  • $v = \frac{dx}{dt} \rightarrow \int_{x_0}^{x} dx = \int_{0}^{t} (v_0 + at) \, dt \rightarrow x - x_0 = v_0 t + \frac{1}{2}at^2$

  • $a = \frac{dv}{dt} = \frac{dv}{dx} \frac{dx}{dt} = v \frac{dv}{dx} \rightarrow \int_{v_0}^{v} v \, dv = \int_{x_0}^{x} a \, dx \rightarrow \frac{1}{2}(v^2 - v_0^2) = a(x - x_0)$

APPLICATIONS AND EXAMPLES

• Example 2.3 (Vertical Motion): A ball is thrown upward at $20\,m\,s^{-1}$ from a building $25.0\,m$ high ($g = 10\,m\,s^{-2}$, taken as $-10$ for upward positive choice).

  • Max height rise: From $v^2 = v_0^2 + 2ay$, $0 = 20^2 + 2(-10)(y - y_0) \rightarrow (y - y_0) = 20\,m$.

  • Total time to hit ground: Using $y = y_0 + v_0 t + \frac{1}{2}at^2$, $0 = 25 + 20t - 5t^2 \rightarrow t = 5\,s$.

• Example 2.4 (Free Fall): Motion under gravity without air resistance ($a = -g = -9.8\,m\,s^{-2}$).

  • Velocity: $v = -9.8t\,m\,s^{-1}$.

  • Distance: $y = -4.9t^2\,m$.

  • Velocity-Distance: $v^2 = -19.6y\,m^2\,s^{-2}$.

• Example 2.5 (Galileo's Law of Odd Numbers): For a body starting from rest under free fall, the distances covered in successive equal intervals of time $\tau$ follow the ratio $1:3:5:7:9:11...$

  • Distance at $t = \tau$ is $y_0 = \frac{1}{2} g \tau^2$.

  • Distance at $t = 2\tau$ is $4y_0$ (change is $3y_0$).

  • Distance at $t = 3\tau$ is $9y_0$ (change is $5y_0$).

• Example 2.6 (Stopping Distance): The distance $d_s$ traveled by a vehicle after applying brakes.

  • Derived from $v^2 = v_0^2 + 2ax$ where $v=0$ and acceleration is $-a$.

  • $d_s = \frac{-v_0^2}{2a}$.

  • Stopping distance is proportional to the square of initial velocity ($d_s \propto v_0^2$).

• Example 2.7 (Reaction Time): The time taken to observe, think, and act.

  • Experiment: Dropping a ruler. If it falls a distance $d$, reaction time $t_r = \sqrt{\frac{2d}{g}}$.

  • For $d = 21.0\,cm$, $t_r \approx 0.2\,s$.

POINTS TO PONDER

• Frame of Reference: Origin and positive direction choice must be specified before assigning signs to displacement, velocity, or acceleration.

• Speeding Up vs. Slowing Down:

  • If velocity and acceleration have the same sign, the object speeds up.

  • If they have opposite signs, the object slows down.

• Sign of Acceleration: Does not inherently indicate if an object is speeding up or slowing down; it depends on the coordinate system.

• Zero Velocity vs. Zero Acceleration: At the peak of a vertical throw, velocity is zero, but acceleration is non-zero ($-g$).

• Continuous Nature: In realistic situations, velocity and acceleration functions are differentiable and smooth; they cannot change abruptly.

EXERCISES AND CONCEPTUAL TASKS

• Point Object Determination: Railway carriages between stations and monkeys on cyclists are often point objects; spinning/tumbling objects near edges often are not.

• Graph Analysis:

  • Slope of $x-t$ graph determines speed ($A$ vs $B$ homes/school).

  • Drunkard's walk ($5$ steps forward, $3$ back) results in net $2\,m$ progress every $8\,s$.

• Retardation Calculation: For a car stopping from $126\,km\,h^{-1}$ ($35\,m\,s^{-1}$) in $200\,m$, retardation $a = -v_0^2 / 2x = -3.06\,m\,s^{-2}$, taking approx $11.4\,s$.

• One-Dimensional Motion Constraints: A particle cannot have two different positions at the same time, nor can it have non-zero velocity with zero magnitude of displacement over time (unless it returns to start).