What is Distance as Integral of Speed? | Simple Explanation for Class 12

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What is Distance Travelled as an Integral of Speed?

Grade Level:

Class 12

AI/ML, Physics, Biotechnology, FinTech, EVs, Space Technology, Climate Science, Blockchain, Medicine, Engineering, Law, Economics

Definition

What is it?

Distance travelled is the total path covered by an object. When an object's speed changes over time, we can find the exact total distance by integrating its speed function over a specific time interval. This means adding up tiny bits of (speed x time) for every moment.

Simple Example

Quick Example

Imagine you are riding your bicycle. If you cycle at a constant speed of 10 km/hr for 2 hours, you cover 10 km/hr * 2 hr = 20 km. But what if your speed changes, like you speed up going downhill and slow down going uphill? To find the total distance, we add up all the tiny distances covered at each changing speed.

Worked Example

Step-by-Step

Let's say a delivery scooter's speed (in km/hr) is given by the function v(t) = 3t^2 + 2t, where 't' is time in hours. We want to find the total distance travelled in the first 2 hours.

1. Identify the speed function: v(t) = 3t^2 + 2t
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2. Identify the time interval: from t = 0 hours to t = 2 hours.
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3. Set up the integral for distance: Distance = Integral from 0 to 2 of v(t) dt = Integral from 0 to 2 of (3t^2 + 2t) dt
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4. Find the antiderivative of the speed function: Antiderivative of (3t^2 + 2t) is (3 * t^3/3 + 2 * t^2/2) = t^3 + t^2
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5. Evaluate the antiderivative at the upper and lower limits: [(2)^3 + (2)^2] - [(0)^3 + (0)^2]
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6. Calculate the values: [8 + 4] - [0 + 0] = 12 - 0 = 12
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7. The total distance travelled is 12 km.

Answer: The scooter travels 12 km in the first 2 hours.

Why It Matters

Understanding this helps engineers design efficient electric vehicles (EVs) by calculating battery usage for varying speeds. It's crucial for ISRO scientists tracking satellites and for AI/ML models predicting traffic flow in smart cities like Bengaluru or Delhi. Knowing this concept opens doors to careers in space technology, automotive engineering, and data science.

Common Mistakes

MISTAKE: Confusing distance with displacement. Students often think distance can be negative or zero even if movement occurred. | CORRECTION: Distance travelled is always positive or zero, as it measures the total path length covered, regardless of direction. Speed is always non-negative.

MISTAKE: Forgetting to include 'dt' in the integral setup or using the wrong variable. | CORRECTION: The 'dt' indicates that we are integrating with respect to time, which is essential for correctly setting up the integral for distance from a speed function.

MISTAKE: Incorrectly applying the limits of integration or making calculation errors during evaluation. | CORRECTION: Always substitute the upper limit first, then the lower limit, and subtract the second result from the first. Double-check all arithmetic.

Practice Questions

Try It Yourself

QUESTION: A robot moves with a speed v(t) = 4t m/s. What is the distance travelled in the first 3 seconds? | ANSWER: 18 meters

QUESTION: A car's speed is given by v(t) = 6t^2 - 2t m/s. Find the distance travelled between t = 1 second and t = 2 seconds. | ANSWER: 10 meters

QUESTION: A drone's speed function is v(t) = 5 - t m/s for 0 <= t <= 5 seconds. Calculate the total distance travelled by the drone in these 5 seconds. | ANSWER: 12.5 meters

MCQ

Quick Quiz

If a particle's speed is v(t) = 2t + 1 m/s, which integral represents the distance travelled from t=0 to t=4 seconds?

Integral from 0 to 4 of (2t + 1) dt

Integral from 0 to 4 of (2t) dt

Integral from 0 to 4 of (2t + 1) dt^2

Integral from 0 to 4 of (1) dt

The Correct Answer Is:

Distance is the integral of speed with respect to time. Option A correctly sets up the integral of the given speed function (2t + 1) over the specified time interval (0 to 4 seconds).

Real World Connection

In the Real World

Imagine a Zomato delivery rider on their scooter. Their speed changes constantly due to traffic, signals, and road conditions. GPS systems and apps like Google Maps use this concept (integrating speed data) to estimate accurate arrival times for your food order, helping them manage deliveries efficiently across Indian cities.

Key Vocabulary

Key Terms

INTEGRAL: A mathematical tool to sum up infinitely many tiny parts | SPEED FUNCTION: A formula that tells us an object's speed at any given time | TIME INTERVAL: The specific start and end times over which we want to calculate distance | ANTIDERIVATIVE: The reverse process of differentiation, used to solve integrals

What's Next

What to Learn Next

Next, you can explore 'Displacement as an Integral of Velocity.' This builds on what you've learned but introduces direction, showing how integrals can track both how far and in which direction an object has moved. Keep practicing, you're doing great!