A-Level Mathematics 9709 - Mechanics (M1)

Kinetics of Motion in a Straight Line

This section covers the fundamental concepts of kinematics, which describe the motion of objects in a straight line. We will explore displacement, velocity, acceleration, and the equations of motion.

1. Displacement, Velocity, and Acceleration

These are the core concepts in describing motion.

• Displacement ($\Delta x$): The change in position of an object. It is a vector quantity, meaning it has both magnitude and direction.
• Average Velocity ($v_{avg}$): The rate of change of displacement over a time interval. It is a vector quantity.
• Instantaneous Velocity ($v$): The velocity of an object at a specific point in time. It is the limit of the average velocity as the time interval approaches zero.
• Average Acceleration ($a_{avg}$): The rate of change of velocity over a time interval. It is a vector quantity.
• Instantaneous Acceleration ($a$): The acceleration of an object at a specific point in time. It is the limit of the average acceleration as the time interval approaches zero.

2. Definitions

Quantity	Symbol	Definition	Units (SI)
Displacement	$\Delta x$	Change in position	m
Average Velocity	$v_{avg}$	$\frac{\Delta x}{\Delta t}$	m/s
Instantaneous Velocity	$v$	$\lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t}$	m/s
Average Acceleration	$a_{avg}$	$\frac{\Delta v}{\Delta t}$	m/s$^2$
Instantaneous Acceleration	$a$	$\lim_{\Delta t \to 0} \frac{\Delta v}{\Delta t}$	m/s$^2$

3. Equations of Motion

These equations relate displacement, velocity, acceleration, and time for objects undergoing constant acceleration in a straight line.

1. $v = u + at$ (Relating final velocity, initial velocity, acceleration, and time)
2. $s = ut + \frac{1}{2}at^2$ (Relating displacement, initial velocity, time, and acceleration)
3. $v^2 = u^2 + 2as$ (Relating final velocity, initial velocity, acceleration, and displacement)
4. $s = \frac{u+v}{2}t$ (Relating displacement, initial velocity, final velocity, and time)

Where:

• $s$ = displacement
• $u$ = initial velocity
• $v$ = final velocity
• $a$ = constant acceleration
• $t$ = time

4. Problem Solving Strategies

To solve problems involving motion in a straight line:

1. Identify the knowns and unknowns.
2. Determine the appropriate equation of motion based on the given information.
3. Ensure consistent units are used.
4. Consider the direction of motion (positive or negative).

5. Example Problem

A car starts from rest and accelerates at a constant rate of 2 m/s$^2$. After 5 seconds, what is its velocity and how far has it travelled?

Solution:

1. Knowns: $u = 0$ m/s, $a = 2$ m/s$^2$, $t = 5$ s
2. Find: $v$ and $s$
3. Use equation 1: $v = u + at = 0 + (2)(5) = 10$ m/s
4. Use equation 2: $s = ut + \frac{1}{2}at^2 = (0)(5) + \frac{1}{2}(2)(5^2) = 25$ m

Therefore, the car's velocity after 5 seconds is 10 m/s and it has travelled 25 meters.