Motion in a Straight Line

Introduction to Motion and the Concept of Point Objects

Motion is an omnipresent phenomenon in the universe, manifesting in various forms such as humans walking or riding bicycles, the flow of blood in arteries, the movement of air in lungs, the falling of leaves, and the rotation of the Earth every twenty-four hours. Even celestial bodies like the Sun and the Milky Way are in constant motion within their respective local groups of galaxies. Motion is formally defined as the change in position of an object over time. In the study of physics, kinematics is the branch that describes motion without investigating its causes, which is the focus of subsequent dynamics chapters. To simplify the description of motion, mathematicians and physicists often utilize the point object approximation. This approximation is considered valid when the size of the object is significantly smaller than the distance it travels over a specific duration. This chapter focuses specifically on rectilinear motion, which is the study of objects moving along a straight line.

Instantaneous Velocity and Speed

While average velocity provides an overview of how fast an object moves over a given time interval, it fails to describe the velocity at specific moments. Instantaneous velocity, or simply velocity $v$, is defined at a specific instant $t$ as the limit of the average velocity as the time interval $\Delta t$ becomes infinitesimally small. Mathematically, this is expressed using the notation of differential calculus as:

$v = \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t} = \frac{dx}{dt}$

Here, $\frac{dx}{dt}$ represents the differential coefficient of position $x$ with respect to time $t$, indicating the rate of change of position at that specific instant. Graphically, the instantaneous velocity at a point $P$ on a position-time graph is equal to the slope of the tangent to the curve at that point. In a numerical analysis of a car's motion modeled by the equation $x = 0.08 t^3$, as the time interval $\Delta t$ centered at $t = 4.0\,s$ is reduced from $2.0\,s$ to $0.01\,s$, the average velocity $\frac{\Delta x}{\Delta t}$ approaches the limiting value of $3.84\,m\,s^{-1}$, which is the instantaneous velocity at that moment.

Instantaneous speed is defined as the magnitude of instantaneous velocity. Unlike average speed, which can be greater than or equal to the magnitude of average velocity over a finite interval, the instantaneous speed at any given moment is always exactly equal to the magnitude of the instantaneous velocity at that same moment. For example, velocities of $+24.0\,m\,s^{-1}$ and $-24.0\,m\,s^{-1}$ both correspond to an instantaneous speed of $24.0\,m\,s^{-1}$.

Understanding Acceleration

Acceleration describes the rate at which the velocity of an object changes over time. Historically, Galileo concluded through his studies of falling objects and inclined planes that the rate of change of velocity with time is a constant for objects in free fall, whereas the change with respect to distance is not constant. Average acceleration $\bar{a}$ over a time interval $\Delta t$ is defined as the change in velocity divided by the time interval:

$\bar{a} = \frac{v_2 - v_1}{t_2 - t_1} = \frac{\Delta v}{\Delta t}$

Instantaneous acceleration $a$ is defined as the limit of the average acceleration as $\Delta t$ approaches zero:

$a = \lim_{\Delta t \to 0} \frac{\Delta v}{\Delta t} = \frac{dv}{dt}$

Graphically, acceleration at an instant is the slope of the tangent to the velocity-time ($v-t$) curve. The SI unit for acceleration is $m\,s^{-2}$. Acceleration can result from changes in speed, changes in direction, or both. It can be positive, negative, or zero. In position-time ($x-t$) graphs, positive acceleration results in a curve that bends upward, negative acceleration results in a curve that bends downward, and zero acceleration results in a straight line. Physical constraints dictate that velocity and acceleration cannot change abruptly; they must be continuous functions.

Kinematic Equations for Uniformly Accelerated Motion

For motion along a straight line with constant (uniform) acceleration, a set of kinematic equations relates displacement ($x$), time ($t$), initial velocity ($v_0$), final velocity ($v$), and acceleration ($a$). For an object starting at $t = 0$ with initial velocity $v_0$, the equations are:

1. $v = v_0 + at$
2. $x = v_0 t + \frac{1}{2} a t^2$
3. $v^2 = v_0^2 + 2ax$

These equations assume the initial position $x_0 = 0$. If the initial position is non-zero, $x$ is replaced by $(x - x_0)$, resulting in:

$x = x_0 + v_0 t + \frac{1}{2} a t^2$$v^2 = v_0^2 + 2a(x - x_0)$

The area under a velocity-time ($v-t$) graph represents the displacement of the object over a specific time interval. For constant acceleration, the displacement is also equal to the product of the average velocity and time:

$x = \bar{v} t = \left( \frac{v + v_0}{2} \right) t$

Calculus Application in Kinematics

The kinematic equations can be derived using calculus, which allows for their application even in cases where acceleration is non-uniform. By integrating $a = \frac{dv}{dt}$, we obtain $v = v_0 + at$. By integrating $v = \frac{dx}{dt}$, we find the expression for position. Furthermore, the relationship between velocity and position can be found using the chain rule:

$a = \frac{dv}{dt} = \frac{dv}{dx} \times \frac{dx}{dt} = v \frac{dv}{dx}$

Integrating $v dv = a dx$ yields the equation $v^2 = v_0^2 + 2a(x - x_0)$.

Motion Under Free Fall and Galileo's Law of Odd Numbers

Free fall occurs when an object is accelerated toward Earth solely under the influence of gravity, neglecting air resistance. Near the Earth's surface, the acceleration due to gravity $g$ is approximately constant at $9.8\,m\,s^{-2}$. Choosing the upward direction as positive, the acceleration in free fall is $a = -g = -9.8\,m\,s^{-2}$. For an object dropped from rest ($v_0 = 0$) at height $y = 0$, the equations of motion simplify to:

$v = -9.8 t$$y = -4.9 t^2$$v^2 = -19.6 y$

Galileo's Law of Odd Numbers states that for an object falling from rest, the distances traversed during successive equal intervals of time follow the ratio of odd numbers: $1:3:5:7:9...$. This can be shown by calculating the position $y = -\frac{1}{2} g t^2$ at intervals $\tau, 2\tau, 3\tau$, etc. The successive distances are found to be $y_0, 3y_0, 5y_0, 7y_0$, and so on, where $y_0 = \frac{1}{2} g \tau^2$.

Stopping Distance and Reaction Time

Stopping distance ($d_s$) is the distance a vehicle travels after brakes are applied before coming to a complete stop. It depends on initial velocity ($v_0$) and the deceleration ($-a$). Setting final velocity $v = 0$ in the kinematic equation $v^2 = v_0^2 + 2ax$, we derive:

$d_s = \frac{-v_0^2}{2a}$

Consequently, the stopping distance is proportional to the square of the initial velocity. Doubling the initial speed quadruples the stopping distance.

Reaction time is the duration between observing a stimulus and taking action. In a ruler-drop experiment where a person catches a ruler falling a distance $d$, the reaction time $t_r$ is calculated using the free fall formula $d = \frac{1}{2} g t_r^2$:

$t_r = \sqrt{\frac{2d}{g}}$

If a ruler falls $21.0\,cm$ ($0.21\,m$) before being caught, the reaction time is approximately $0.2\,s$.

Points to Ponder and Graphical Interpretation

1. Sign Convention: The choice of origin and positive direction is arbitrary but must be consistent. Quantities like displacement, velocity, and acceleration are algebraic and can be positive or negative.
2. Speeding Up vs. Slowing Down: If acceleration is in the same direction as velocity, the object speeds up. If acceleration is in the opposite direction, the object slows down. The sign of acceleration alone does not indicate if an object is speeding up; for instance, a falling object has negative acceleration (downward) and negative velocity (downward), resulting in an increase in speed.
3. Zero Velocity: An object can have zero velocity momentarily but still have non-zero acceleration. A prime example is an object at the peak of its trajectory when thrown vertically upward; its velocity is zero, but its acceleration remains $g$.
4. Graph Characteristics: For uniform motion, the $x-t$ graph is a straight inclined line and acceleration is zero. For uniform acceleration, the $x-t$ graph is a parabola, while the $v-t$ graph is a straight inclined line.

Discussion and Practical Scenarios

In the exercises provided in the text, several complex scenarios are discussed to test the application of these principles:

• Point Object Status: A railway carriage moving between stations can be considered a point object due to the large distance compared to its size, whereas a tumbling beaker or a sharply turning cricket ball may not be.
• Comparison of Motion: In examining $x-t$ graphs for two people (e.g., Children A and B returning from school), comparing the slopes identifies who walks faster, while comparing the intercepts on the position axis identifies who lives closer to the fixed origin.
• Relative Velocity and Collisions: Scenarios like a police van ($30\,km\,h^{-1}$) firing a bullet (muzzle speed $150\,m\,s^{-1}$) at a thief's car ($192\,km\,h^{-1}$) require converting all speeds to the same units and considering the relative velocity of the bullet with respect to the target car to determine the impact speed.