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What is Acceleration as the Second Derivative of Displacement?
Grade Level:
Class 12
AI/ML, Physics, Biotechnology, FinTech, EVs, Space Technology, Climate Science, Blockchain, Medicine, Engineering, Law, Economics
Definition
What is it?
Acceleration as the second derivative of displacement means how fast the velocity changes over time. If you know the equation for an object's position (displacement) over time, you can find its acceleration by differentiating that equation twice.
Simple Example
Quick Example
Imagine you are driving an auto-rickshaw. Your position (how far you've gone) changes over time. The rate at which your position changes is your speed (velocity). The rate at which your speed changes (like when you press the accelerator or brake) is your acceleration. Taking the derivative twice tells you this change in speed.
Worked Example
Step-by-Step
Let's say a car's displacement (position) from its starting point is given by the equation: x(t) = 3t^2 + 2t + 5, where x is in meters and t is in seconds.
---Step 1: Find the first derivative to get velocity. Velocity (v) is dx/dt.
v(t) = d/dt (3t^2 + 2t + 5)
v(t) = 6t + 2
---Step 2: Find the second derivative to get acceleration. Acceleration (a) is dv/dt or d^2x/dt^2.
a(t) = d/dt (6t + 2)
a(t) = 6
---Step 3: The acceleration is a constant 6 meters per second squared. This means the car's velocity is increasing by 6 m/s every second.
Answer: The acceleration of the car is 6 m/s^2.
Why It Matters
Understanding this concept is key for designing safe cars and high-speed trains (EVs), predicting satellite orbits (Space Technology), and even creating realistic animations in video games. Engineers use this daily to make sure things move exactly as planned, impacting careers in robotics and aerospace.
Common Mistakes
MISTAKE: Confusing velocity with acceleration | CORRECTION: Velocity is the first derivative of displacement (how fast position changes). Acceleration is the second derivative (how fast velocity changes).
MISTAKE: Forgetting to differentiate constants to zero | CORRECTION: When differentiating, any constant term (like the '5' in 3t^2 + 2t + 5) becomes zero after the first differentiation because its rate of change is zero.
MISTAKE: Mixing up units for displacement, velocity, and acceleration | CORRECTION: Displacement is in meters (m), velocity in meters per second (m/s), and acceleration in meters per second squared (m/s^2). Always check and write the correct units.
Practice Questions
Try It Yourself
QUESTION: If the displacement of a ball is given by x(t) = 4t^3 + t, what is its acceleration? | ANSWER: a(t) = 24t
QUESTION: A drone's position is described by x(t) = 2t^2 - 5t + 10. What is its acceleration at t = 3 seconds? | ANSWER: a(t) = 4 m/s^2. (The acceleration is constant, so it's 4 m/s^2 at any time t).
QUESTION: The displacement of a bullet is given by x(t) = 5t^4 - 2t^3 + 7t - 1. Find the acceleration at t = 1 second. | ANSWER: a(t) = 60t^2 - 12t. At t=1, a(1) = 60(1)^2 - 12(1) = 48 m/s^2.
MCQ
Quick Quiz
If the displacement of an object is given by s(t) = 5t^2 + 3t, what is its acceleration?
10t + 3
5t + 3
10
5
The Correct Answer Is:
C
The first derivative (velocity) is 10t + 3. The second derivative (acceleration) is 10. Options A and B are velocity-related, and D is incorrect.
Real World Connection
In the Real World
When you book a ride on an app like Ola or Uber, the app's navigation system constantly calculates your car's position, velocity, and acceleration. This helps it estimate your arrival time, detect sudden stops or accelerations for safety, and even manage traffic flow in smart city systems.
Key Vocabulary
Key Terms
DISPLACEMENT: An object's change in position. | VELOCITY: The rate of change of displacement. | ACCELERATION: The rate of change of velocity. | DERIVATIVE: A mathematical tool to find the rate of change of a function. | FUNCTION: A rule that assigns each input exactly one output.
What's Next
What to Learn Next
Now that you understand acceleration, you can explore 'Jerk as the Third Derivative of Displacement.' This will show you how acceleration itself changes, which is important in advanced physics and engineering for studying sudden impacts and smooth motion.
