Problem 7 - Master Resource Book in JEE Main Physics

Question

A bird flies for \(4 \mathrm{~s}\) with a velocity of \(|t-2| \mathrm{ms}^{-1}\) in a straight line, where \(t=\) time in second. It covers a distance of (a) \(8 \mathrm{~m}\) (b) \(6 \mathrm{~m}\) (c) \(4 \mathrm{~m}\) (d) \(2 \mathrm{~m}\)

Step-by-Step Solution

Answer

The bird covers a distance of 4 m in 4 seconds.

Step 1: Understanding the Problem

To find the total distance covered by the bird in 4 seconds, we need to integrate the velocity function \(|t-2|\) over the interval from \(t = 0\) to \(t = 4\).

Step 2: Analyze the Velocity Function

The velocity function \(|t-2|\) changes depending on the value of \(t\). If \(t < 2\), \(|t-2| = 2-t\). If \(t \geq 2\), \(|t-2| = t-2\). This implies that we need to split the interval [0, 4] into two parts: [0, 2] and [2, 4].

Step 3: Integrate the First Interval

In the interval \(t \in [0, 2]\), the velocity function is \(2-t\). We calculate the integral: \[\int_{0}^{2} (2-t) \, dt = \left[2t - \frac{t^2}{2}\right]_{0}^{2} = (4 - 2) - (0 - 0) = 2\mathrm{~m}.\]

Step 4: Integrate the Second Interval

In the interval \(t \in [2, 4]\), the velocity function is \(t-2\). We calculate the integral: \[\int_{2}^{4} (t-2) \, dt = \left[\frac{t^2}{2} - 2t\right]_{2}^{4} = \left[8 - 8\right] - [2 - 4] = 2\mathrm{~m}.\]

Step 5: Calculate Total Distance Covered

Add the distances covered in the two intervals to get the total distance: \[2 \mathrm{~m} (from\ [0, 2]) + 2 \mathrm{~m} (from\ [2, 4]) = 4 \mathrm{~m}.\] This is the total distance covered by the bird in 4 seconds.

Key Concepts

Velocity function

In physics, the velocity function gives us insight into how an object's speed and direction change over time. For the bird in our exercise, the velocity function is given as \(|t-2|\). This equation is expressed in terms of "absolute value," which means that it reflects only non-negative numbers.

Here's a simple breakdown of what the velocity function signifies:

• If the time \(t\) is less than 2, the output from the function is \(2-t\). This forms a linear function that decreases as time progresses within that interval.
• When \(t\) is equal to or greater than 2, the function becomes \(t-2\), indicating a different type of motion during this period.

Understanding the construction of a velocity function can help us predict and analyze the movement of objects within specific time frames.

Definite integrals

When dealing with physics problems like determining the distance the bird travels, definite integrals are very helpful. Definite integrals allow us to calculate the accumulation of quantities, such as distance, over a given interval.

The process involves integrating the velocity function over specific time intervals. Each of these intervals provides a portion of the total distance based on the behavior of the velocity function:

• The first integral \(\int_{0}^{2} (2-t) \, dt\) evaluates how far the bird travels when \(t\) is between 0 to 2.
• The second integral \(\int_{2}^{4} (t-2) \, dt\) evaluates the distance covered during \(t\) being 2 to 4.

By adding these definite integrals, we receive a complete measure of the total distance traveled over the specified time frame, delivering meaningful insight into the bird's journey.

Kinematics in one dimension

Kinematics in one dimension involves studying the motion of objects along a single straight line, which simplifies calculations. In our scenario involving a bird's flight, kinematics focuses on parameters like velocity and distance to understand motion over time.

Here’s why one-dimensional kinematics is straightforward:

• The motion is restricted to one line, simplifying translations to calculations using time-dependent functions like velocity.
• In our problem, since the bird moves in a straight line, we apply concepts like distance and velocity to determine overall movement without considering other dimensions.

This makes it easier to model and solve problems effectively by applying kinematic equations and integral calculus. Through such rigorous application, we gain clarity on the complete dynamics of objects, even with complex underlying functions like \(|t - 2|\).