The first equation of motion is a fundamental concept in physics, particularly in kinematics. It relates the initial velocity, final velocity, acceleration, and time of an object in motion. The equation is expressed as:
v = u + at
Where:
- = Final velocity (m/s)
- = Initial velocity (m/s)
- = Acceleration (m/s²)
- = Time (s)
The graphical method provides a clear and intuitive understanding of this equation. This topic will explain the derivation of the first equation of motion using velocity-time graphs, ensuring clarity and accuracy.
Understanding the Velocity-Time Graph
What Is a Velocity-Time Graph?
A velocity-time graph shows how an object’s velocity changes with time. The:
- X-axis (horizontal axis) represents time (t).
- Y-axis (vertical axis) represents velocity (v).
For uniform acceleration, the graph is a straight line. The slope of this line represents the acceleration of the object.
Key Elements of the Graph
- Initial Velocity (u): The velocity of the object at time $t = 0$ .
- Final Velocity (v): The velocity after time .
- Acceleration (a): The rate of change of velocity with respect to time.
- Time (t): The duration for which the object is in motion.
Derivation of the First Equation of Motion by Graphical Method
Step 1: Drawing the Velocity-Time Graph
- Draw a graph with time on the X-axis and velocity on the Y-axis.
- Plot the initial velocity at time $t = 0$ .
- From , draw a straight line sloping upwards if the object accelerates uniformly.
- The final velocity is reached at time .
This forms a right-angled trapezium on the graph.
Step 2: Understanding the Graph Components
- The vertical side represents the change in velocity ( $v – u$ ).
- The horizontal side represents the time ( ).
- The slope of the graph gives the acceleration ( ).
Mathematically, the slope is:
a = frac{v – u}{t}
Step 3: Deriving the Equation
By rearranging the above formula:
v – u = at
v = u + at
This is the first equation of motion, derived purely using the graphical method.
Interpretation of the First Equation of Motion
What Does the Equation Mean?
The equation $v = u + at$ shows that:
- The final velocity of an object depends on its initial velocity, the acceleration, and the time for which it accelerates.
- If acceleration is zero, the final velocity equals the initial velocity.
- If initial velocity is zero (object starts from rest), the equation simplifies to $v = at$ .
Real-Life Applications
- Vehicles Accelerating on Roads: Calculating the final speed after a certain time.
- Athletes in a Race: Estimating how fast a runner reaches top speed.
- Free-Falling Objects: Determining the velocity just before hitting the ground (assuming uniform acceleration due to gravity).
Graphical Insights into the Equation
Slope as Acceleration
The steeper the slope, the greater the acceleration. A horizontal line would indicate zero acceleration, meaning the object moves with constant velocity.
Area Under the Curve
While the first equation focuses on velocity and acceleration, the area under the velocity-time graph represents the displacement of the object. This links to other equations of motion.
Importance of the Graphical Method
1. Visual Understanding
Graphs provide a visual interpretation of how velocity changes over time, making it easier for learners to grasp the relationships between physical quantities.
2. Foundation for Advanced Concepts
The graphical method builds a strong foundation for understanding more complex kinematic equations and motion analysis.
3. Error Minimization
Graphical approaches reduce calculation errors, especially when dealing with real-world data that can be represented visually for quick interpretation.
Common Mistakes to Avoid
- Confusing Slope with Speed: The slope of the velocity-time graph represents acceleration, not speed.
- Incorrect Unit Usage: Ensure velocity is in m/s, time in seconds (s), and acceleration in m/s².
- Assuming Non-Linear Graphs: The derivation assumes uniform acceleration. For non-uniform acceleration, the graph would not be a straight line.
Practice Problems
Problem 1
A car starts from rest and accelerates uniformly at $4 , m/s²$ for seconds. Find its final velocity.
Solution:
Given:
- $u = 0 , m/s$
- $a = 4 , m/s²$
- $t = 5 , s$
Using $v = u + at$ :
v = 0 + (4 times 5) = 20 , m/s
Final velocity = 20 m/s.
Problem 2
A train moves with an initial velocity of $15 , m/s$ and accelerates uniformly at $2 , m/s²$ for seconds. What is its final velocity?
Solution:
Given:
- $u = 15 , m/s$
- $a = 2 , m/s²$
- $t = 10 , s$
Using $v = u + at$ :
v = 15 + (2 times 10) = 35 , m/s
Final velocity = 35 m/s.
The first equation of motion ( $v = u + at$ ) is a powerful tool for understanding linear motion under uniform acceleration. By using the graphical method, learners gain a clear, visual insight into how velocity, time, and acceleration interrelate.
This graphical approach not only simplifies complex calculations but also lays the groundwork for more advanced studies in physics. With a solid grasp of this concept, students can confidently explore further kinematic equations and real-world motion scenarios.