Linear Motion: Principles and Worked Examples

Fundamentals of Linear Motion

Linear motion refers to the movement of objects in one dimension along a straight path. This includes horizontal motion, vertical motion, and motion on inclined surfaces. Kinematics is the specific study of motion without regarding the forces that cause it, whereas dynamics focuses on those causes.

Key Variables and Physical Quantities

• Distance: Defined as the change in position relative to a reference point. It is a scalar quantity, meaning it has magnitude only. It is measured in metres ($m$) and is always positive.
• Displacement: Defined as the change in position relative to a reference point in a specific direction. It is a vector quantity, possessing both magnitude and direction. It is measured in metres ($m$) and can be positive or negative depending on the direction. Common symbols include $s$, $x$, and $y$.
• Average Speed: The rate of change of distance. It is a scalar quantity measured in metres per second ($m/s$ or $ms^{-1}$).
  • Mathematical Formula: $\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}}$
• Instantaneous Speed: The speed at a specific point in time, such as that shown on a car's speedometer. Mathematically, it is the derivative of distance with respect to time: $v = \frac{dx}{dt}$.
• Average Velocity: The rate of change of displacement. It is a vector quantity measured in metres per second ($m/s$).
  • Mathematical Formula: $\text{Average Velocity} = \frac{\text{Displacement}}{\text{Total Time}}$
• Instantaneous Velocity: Velocity at a specific moment, calculated by reducing the change in time to nearly zero: $v = \frac{ds}{dt}$. Velocity is alternatively defined as speed in a particular direction.
• Average Acceleration: The rate of change of velocity. It is a vector quantity measured in metres per second squared ($m/s^2$ or $ms^{-2}$).
  • Mathematical Formula: $a = \frac{v - u}{t}$, where $v$ is final velocity, $u$ is initial velocity, and $t$ is time.
• Instantaneous Acceleration: Acceleration at a specific moment: $a = \frac{dv}{dt}$.

Critical Definitions and Distinctions

• Sign Conventions: If velocity increases, velocity and acceleration must have the same sign (both positive or both negative). If velocity decreases, their signs must be opposite.
• Acceleration Causes: Acceleration produced by a change in the magnitude of velocity, the direction of velocity, or both. For example, a body in a circular path at constant speed is still accelerating because its direction changes continuously.
• Deceleration vs. Retardation: Negative acceleration is termed deceleration. Retardation specifically describes a body slowing down.
• Particle: A term used in mechanics to refer to point-like matter with negligible size and mass.
• Relation of Average and Instantaneous Values: When acceleration or velocity is constant, the average and instantaneous values are equal.

Equations of Motion for Constant Acceleration

The following equations are valid only when acceleration is uniform (constant).

General Equations of Motion

1. $v = u + at$
2. $s = ut + \frac{1}{2}at^2$
3. $v^2 = u^2 + 2as$
4. $s = \frac{1}{2}(u + v)t$
5. $s = vt - \frac{1}{2}at^2$

Equations Adjusted for Gravity

For vertical motion, the general equations are modified by substituting acceleration ($a$) with the acceleration due to gravity ($g$) and distance ($s$) with height ($h$).

1. $v = u + gt$
2. $h = ut + \frac{1}{2}gt^2$
3. $v^2 = u^2 + 2gh$
4. $h = \frac{1}{2}(u + v)t$
5. $h = vt - \frac{1}{2}gt^2$

• Value of g: Correct to two significant figures, $g$ is approximately $9.8\,m/s^2$ at sea level. For quick calculations, $10\,m/s^2$ is often used. When an object falls downward toward Earth, $g$ is positive. When moving upward, $g$ is negative ($-9.8\,m/s^2$).
• Note on Equation 5: This equation is derived by substituting $u = v - at$ (from Equation 1) into Equation 2.

Linear Motion Graphs

Graphs are used to solve complex motion problems where equations alone might be difficult. Time ($t$) is usually plotted on the horizontal ($x$) axis.

Types of Graphs and Interpretations

1. Distance-Time Graph: The slope (gradient) represents speed. The area under the graph is meaningless.
2. Displacement-Time Graph: The slope represents velocity. The area under the graph is meaningless.
3. Speed-Time Graph: The slope is meaningless (in this specific context), but the area under the graph represents distance.
4. Velocity-Time Graph: The slope represents acceleration. The area under the graph represents displacement.
5. Acceleration-Time Graph: The slope is meaningless, but the area under the graph represents velocity.

Summary Table of Calculus Relations

Advanced Scenarios and Calculations

Multi-Stage Journey Analysis

In journeys with varying speeds, calculating average speed requires finding the total distance and total time for all segments combined.

• Example Case: A plane flies $5500\,km$ at $800\,km/h$ and returns at $900\,km/h$.
• $t_1 = \frac{5500}{800} = 6.875\,h$
• $t_2 = \frac{5500}{900} = 6.111\,h$
• $\text{Average Speed} = \frac{5500 + 5500}{6.875 + 6.111} = 847\,km/h$

Reaction Time (RT) and Braking

The total distance traveled before a vehicle stops includes the distance covered during the driver's reaction time (at constant velocity) and the distance covered during braking (at uniform retardation).

• Influence of Alcohol: A drunk driver's reaction time might be $2.0\,s$ compared to a normal driver's $0.7\,s$. At a velocity of $40\,m/s$, this results in identifying an extra distance of $d = v \times (RT_2 - RT_1) = 40 \times (2.0 - 0.7) = 52\,m$.

Vertical Motion under Gravity (Free Fall)

• Maximum Height: At the maximum height of a vertical projection, the final velocity ($v$) is zero.
• Time of Flight: In the absence of air resistance, the time taken to rise to the maximum height is equal to the time taken to fall back to the original projection point. The total return time is $t_{\text{total}} = \frac{2u}{g}$.
• Height of Buildings/Cliffs: Using $h = ut + \frac{1}{2}gt^2$, the height of a building can be determined if the time for an object to fall from rest is known ($h = 4.9t^2$ when using $g=9.8$).
• Multi-Storey Intervals: The distance between floors in a building can be solved by analyzing the time taken for a falling object to pass specific floors. If a ball takes $0.15\,s$ to fall between the 15th and 14th floors, and $0.12\,s$ between the 14th and 13th, the initial velocity at the 15th floor can be solved using simultaneous equations of the form $s = ut + \frac{1}{2}gt^2$.

Simultaneous/Chase Motion

Two objects meet when their displacements from a common origin are equal ($s_1 = s_2$). This is analyzed by setting their motion equations equal to each other to solve for time ($t$) and then substituting back to find the distance.

• Example Train Meet: Train 1 ($T_1$) and Train 2 ($T_2$) starting from terminals $40\,km$ apart. $\frac{1}{2}a_1t_1^2 = 40000 - \frac{1}{2}a_2t_2^2$ (assuming they move toward each other).