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What is Displacement as a Definite Integral of Velocity?
Grade Level:
Class 12
AI/ML, Physics, Biotechnology, FinTech, EVs, Space Technology, Climate Science, Blockchain, Medicine, Engineering, Law, Economics
Definition
What is it?
Displacement as a definite integral of velocity means finding the total change in an object's position by adding up all the tiny movements it makes over a specific time period. It's like calculating how far you've actually moved from your starting point, regardless of the path taken, by knowing your speed and direction at every moment.
Simple Example
Quick Example
Imagine an auto-rickshaw driver in Delhi. If you know the auto's speed and direction at every second from the moment it leaves the stand till it reaches your school, you can use integration to find the exact straight-line distance (displacement) from the stand to your school, even if the auto took many turns and stops.
Worked Example
Step-by-Step
Let's say a toy car's velocity (v) is given by the function v(t) = 3t^2 - 2t meters per second, where 't' is time in seconds. We want to find the displacement of the car from t = 1 second to t = 3 seconds.
Step 1: Understand that displacement is the definite integral of velocity over a time interval. So, Displacement = ∫v(t) dt from t=1 to t=3.
---Step 2: Write down the integral: Displacement = ∫(3t^2 - 2t) dt from 1 to 3.
---Step 3: Find the antiderivative of the velocity function. The antiderivative of 3t^2 is (3t^3)/3 = t^3. The antiderivative of -2t is (-2t^2)/2 = -t^2.
---Step 4: So, the antiderivative is F(t) = t^3 - t^2.
---Step 5: Now, evaluate F(t) at the upper limit (t=3) and the lower limit (t=1).
---Step 6: F(3) = (3)^3 - (3)^2 = 27 - 9 = 18.
---Step 7: F(1) = (1)^3 - (1)^2 = 1 - 1 = 0.
---Step 8: Subtract the value at the lower limit from the value at the upper limit: Displacement = F(3) - F(1) = 18 - 0 = 18 meters.
Answer: The displacement of the toy car from t=1 second to t=3 seconds is 18 meters.
Why It Matters
This concept is crucial for designing rockets in Space Technology, predicting vehicle movement in EVs, and even understanding fluid flow in Biotechnology. Engineers use it to calculate how far a robot arm moves or how much material flows through a pipe, opening doors to careers in robotics and aerospace.
Common Mistakes
MISTAKE: Confusing displacement with total distance traveled. | CORRECTION: Displacement is the net change in position (straight line from start to end), while total distance considers the entire path, even if you go back and forth. Displacement can be negative or zero, total distance is always positive.
MISTAKE: Forgetting to evaluate the antiderivative at both the upper and lower limits. | CORRECTION: Always subtract F(lower limit) from F(upper limit) to find the definite integral's value.
MISTAKE: Not correctly finding the antiderivative of the velocity function. | CORRECTION: Practice your integration rules (power rule, etc.) carefully. Remember that ∫t^n dt = (t^(n+1))/(n+1) + C.
Practice Questions
Try It Yourself
QUESTION: A particle's velocity is given by v(t) = 4t - 1 meters/second. Find its displacement from t = 0 to t = 2 seconds. | ANSWER: 6 meters
QUESTION: If a bike's velocity is v(t) = t^2 + 2t + 3 meters/second, what is its displacement from t = 1 to t = 3 seconds? | ANSWER: 28.67 meters (or 86/3 meters)
QUESTION: A drone's velocity function is v(t) = sin(pi*t) meters/second. Calculate its displacement from t = 0 to t = 1 second. (Hint: ∫sin(ax) dx = -1/a * cos(ax)) | ANSWER: 2/pi meters
MCQ
Quick Quiz
If the velocity of an object is given by v(t) = 2t, what is its displacement from t=0 to t=4 seconds?
4 meters
8 meters
16 meters
32 meters
The Correct Answer Is:
C
The integral of 2t from 0 to 4 is [t^2] from 0 to 4. This equals (4^2) - (0^2) = 16 - 0 = 16 meters.
Real World Connection
In the Real World
In India, ISRO scientists use similar integral calculations to precisely track the displacement of satellites and rockets, ensuring they reach their target orbits. Delivery apps like Zepto or Swiggy use algorithms that, at their core, involve understanding displacement to optimize delivery routes and predict arrival times, even though they account for traffic and road networks.
Key Vocabulary
Key Terms
VELOCITY: The rate of change of an object's position with respect to time, including its speed and direction. | DISPLACEMENT: The overall change in an object's position from its starting point to its ending point, a vector quantity. | DEFINITE INTEGRAL: An integral with upper and lower limits, representing the area under a curve between those limits. | ANTIDERIVATIVE: The reverse process of differentiation; a function whose derivative is the original function. | LIMITS OF INTEGRATION: The specific start and end points of the interval over which the integral is calculated.
What's Next
What to Learn Next
Next, you can explore how to calculate the total distance traveled using the definite integral of the absolute value of velocity. This builds on your understanding of displacement and helps you solve more complex real-world motion problems.
