Introduction
Motion in a straight line, also known as rectilinear motion, is one of the fundamental concepts in physics and is crucial for understanding more complex topics. This study note will cover the essential aspects of motion in a straight line, including displacement, velocity, acceleration, and the equations of motion. We'll break down each concept, provide examples, and highlight important points to help you grasp the material effectively.
Displacement
Displacement is a vector quantity that refers to the change in position of an object. It has both magnitude and direction.
Definition
Displacement is defined as the shortest distance from the initial to the final position of an object. It is represented by the symbol $\vec{d}$.
Formula
If an object moves from position $x_i$ to $x_f$, the displacement $\vec{d}$ is given by:
$$ \vec{d} = x_f - x_i $$
ExampleAn object moves from position $x_i = 2 , \text{m}$ to $x_f = 5 , \text{m}$. The displacement is:
$$ \vec{d} = 5 , \text{m} - 2 , \text{m} = 3 , \text{m} $$
NoteDisplacement is different from distance, which is a scalar quantity and only considers the magnitude of the path traveled.
Velocity
Velocity is a vector quantity that describes the rate of change of displacement with respect to time.
Average Velocity
The average velocity $\vec{v}_{avg}$ is defined as the total displacement divided by the total time taken.
$$ \vec{v}_{avg} = \frac{\vec{d}}{t} $$
Instantaneous Velocity
Instantaneous velocity is the velocity of an object at a specific moment in time. It is given by the derivative of displacement with respect to time:
$$ \vec{v} = \frac{d\vec{d}}{dt} $$
ExampleIf the displacement of an object is given by $x(t) = 5t^2$, the instantaneous velocity at time $t$ is:
$$ \vec{v} = \frac{d(5t^2)}{dt} = 10t $$
TipRemember that velocity has both magnitude and direction. A positive velocity indicates motion in the positive direction, while a negative velocity indicates motion in the negative direction.
Acceleration
Acceleration is a vector quantity that describes the rate of change of velocity with respect to time.
Average Acceleration
The average acceleration $\vec{a}_{avg}$ is defined as the change in velocity divided by the total time taken.
$$ \vec{a}_{avg} = \frac{\Delta \vec{v}}{t} $$
Instantaneous Acceleration
Instantaneous acceleration is the acceleration of an object at a specific moment in time. It is given by the derivative of velocity with respect to time:
$$ \vec{a} = \frac{d\vec{v}}{dt} $$
ExampleIf the velocity of an object is given by $v(t) = 10t$, the instantaneous acceleration at time $t$ is:
$$ \vec{a} = \frac{d(10t)}{dt} = 10 , \text{m/s}^2 $$
NoteAcceleration can be positive (speeding up) or negative (slowing down), also known as deceleration.
Equations of Motion
The equations of motion describe the relationship between displacement, velocity, acceleration, and time. They are applicable when acceleration is constant.
First Equation of Motion
$$ v = u + at $$
Where:
- $v$ = final velocity
- $u$ = initial velocity
- $a$ = acceleration
- $t$ = time
Second Equation of Motion
$$ s = ut + \frac{1}{2}at^2 $$
Where:
- $s$ = displacement
- $u$ = initial velocity
- $a$ = acceleration
- $t$ = time
Third Equation of Motion
$$ v^2 = u^2 + 2as $$
Where:
- $v$ = final velocity
- $u$ = initial velocity
- $a$ = acceleration
- $s$ = displacement
An object starts from rest ($u = 0$) and accelerates at $2 , \text{m/s}^2$ for $5$ seconds. Find the final velocity and displacement.
Using the first equation of motion:
$$ v = u + at = 0 + (2 , \text{m/s}^2)(5 , \text{s}) = 10 , \text{m/s} $$
Using the second equation of motion:
$$ s = ut + \frac{1}{2}at^2 = 0 + \frac{1}{2}(2 , \text{m/s}^2)(5 , \text{s})^2 = 25 , \text{m} $$
Common MistakeOne common mistake is confusing displacement with distance. Remember that displacement considers direction, while distance does not.
Graphical Representation
Position-Time Graph
A position-time graph shows the position of an object over time. The slope of the graph gives the velocity.
Velocity-Time Graph
A velocity-time graph shows the velocity of an object over time. The slope of the graph gives the acceleration, and the area under the graph gives the displacement.
TipPractice drawing and interpreting these graphs as they are frequently tested concepts.
Conclusion
Understanding motion in a straight line is fundamental for mastering physics. By breaking down displacement, velocity, acceleration, and the equations of motion, you can better understand how objects move. Practice problems and real-world examples will help solidify these concepts.
