What is the Third Equation of Motion?

S6-SA3-0015

Grade Level:

Class 10

AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine

Definition

What is it?

The Third Equation of Motion describes the relationship between an object's final velocity, initial velocity, acceleration, and the distance it travels, without directly involving time. It's a powerful tool when you know how fast something starts, how fast it ends, and how quickly its speed changes, but don't know the exact time taken.

Simple Example

Quick Example

Imagine a delivery scooter starts from a signal at 10 m/s and speeds up to 20 m/s over a certain distance. If we know the scooter's acceleration, the third equation helps us find out exactly how much distance it covered during this speed change, without needing to calculate the time taken.

Worked Example

Step-by-Step

PROBLEM: A car accelerates uniformly from an initial speed of 15 m/s to a final speed of 25 m/s. If its acceleration is 2 m/s^2, what distance does the car cover during this acceleration?

STEP 1: Identify the known values.
Initial velocity (u) = 15 m/s
Final velocity (v) = 25 m/s
Acceleration (a) = 2 m/s^2
Distance (s) = ?

---STEP 2: Write down the Third Equation of Motion.
v^2 = u^2 + 2as

---STEP 3: Substitute the known values into the equation.
(25)^2 = (15)^2 + 2 * (2) * s

---STEP 4: Calculate the squares.
625 = 225 + 4s

---STEP 5: Rearrange the equation to solve for 's'.
625 - 225 = 4s
400 = 4s

---STEP 6: Solve for 's'.
s = 400 / 4
s = 100 meters

ANSWER: The car covers a distance of 100 meters.

Why It Matters

This equation is crucial for engineers designing vehicles and roller coasters, ensuring they stop safely or reach desired speeds over specific distances. In space technology, ISRO scientists use it to calculate rocket trajectories and satellite orbits. Even in AI/ML, understanding these physics principles helps in simulating object movements for robotics and computer vision applications.

Common Mistakes

MISTAKE: Forgetting to square the velocities (v and u). | CORRECTION: Always remember that 'v' and 'u' are squared in this equation (v^2 and u^2).

MISTAKE: Mixing up units, e.g., using km/h for velocity and m/s^2 for acceleration. | CORRECTION: Ensure all quantities are in consistent units (e.g., meters, seconds, m/s, m/s^2) before solving.

MISTAKE: Incorrectly rearranging the equation, especially when 'a' or 's' is negative (deceleration). | CORRECTION: Treat 'a' as negative for deceleration. Carefully isolate the variable you're solving for using proper algebraic steps.

Practice Questions

Try It Yourself

QUESTION: A bicycle moving at 5 m/s accelerates at 1 m/s^2 over a distance of 12 meters. What is its final velocity? | ANSWER: 7 m/s

QUESTION: A train traveling at 72 km/h applies brakes and decelerates uniformly at 2 m/s^2. If it comes to a complete stop, what distance does it cover during braking? (Hint: First convert km/h to m/s). | ANSWER: 100 meters

QUESTION: An object starts from rest and achieves a velocity of 30 m/s after covering a distance of 90 meters. What is its acceleration? | ANSWER: 5 m/s^2

MCQ

Quick Quiz

Which of the following equations relates final velocity, initial velocity, acceleration, and displacement without involving time?

v = u + at

s = ut + (1/2)at^2

v^2 = u^2 + 2as

s = (v+u)/2 * t

The Correct Answer Is:

Option C, v^2 = u^2 + 2as, is the Third Equation of Motion, which connects final velocity, initial velocity, acceleration, and displacement without needing time. The other options are the first, second, and average velocity equations, respectively, all of which include time.

Real World Connection

In the Real World

When you see a metro train accelerating from one station and then decelerating to stop at the next, the engineers use the Third Equation of Motion to design the track length and braking systems. Similarly, when a cricketer hits a six, physicists can estimate how far the ball travels using its initial speed, final speed, and acceleration (due to gravity and air resistance).

Key Vocabulary

Key Terms

VELOCITY: The speed of an object in a given direction. | ACCELERATION: The rate at which an object's velocity changes. | DISPLACEMENT: The overall change in an object's position from its starting point. | INITIAL VELOCITY: The velocity of an object at the beginning of its motion. | FINAL VELOCITY: The velocity of an object at the end of its motion.

What's Next

What to Learn Next

Now that you understand the Third Equation of Motion, explore how to apply all three equations of motion together to solve more complex problems involving objects under gravity. This will prepare you for advanced topics in kinematics and dynamics!