Problem 5 - Master Resource Book in JEE Main Physics

Question

The displacement of a body along \(x\)-axis depends on time as \(\sqrt{x}=t+1\). Then, the velocity of body (a) increase with time (b) decrease with time (c) independent of time (d) None of these

Step-by-Step Solution

Answer

The velocity of the body increases with time (option a).

Step 1: Understand the Problem

We are given that the displacement related to time as \( \sqrt{x} = t + 1 \). We need to find out how velocity, \( v = \frac{dx}{dt} \), changes with time.

Step 2: Express Displacement in Terms of Time

The given equation is \( \sqrt{x} = t + 1 \). First, square both sides to get \( x = (t + 1)^2 \). This expresses displacement \( x \) as a function of time \( t \).

Step 3: Differentiate to Find Velocity

To find velocity, differentiate displacement with respect to time. Start with \( x = (t + 1)^2 \). Using the power rule, differentiate to get:\[ \frac{dx}{dt} = 2(t + 1) \] which represents the velocity \( v \).

Step 4: Analyze the Velocity Expression

The expression for velocity is \( v = 2(t + 1) \). Analyzing this, observe that as \( t \) increases, velocity also increases linearly with time.

Step 5: Conclude Based on Analysis

Since the derived expression for velocity \( v = 2(t + 1) \) is increasing with time, we conclude that the correct answer is option (a): the velocity increases with time.

Key Concepts

motion along x-axis

Understanding motion along the x-axis involves examining how an object moves horizontally over time. In this particular exercise, a particle's motion is described by its displacement along the x-axis. Displacement is a crucial concept, as it shows the change in position of a particle over time, mentioned here by the equation \( \sqrt{x} = t + 1 \).
This equation means that as time \( t \) progresses, the particle's position \( x \) along the axis changes according to the square of \( t+1 \). Motion along a straight line simplifies our calculations and aids in understanding basic kinematic concepts before introducing additional complexities such as curved paths or varying accelerations.

differentiation in physics

Differentiation in physics serves as a powerful tool to uncover the rate at which a quantity changes. In this scenario, we're interested in finding how the velocity of the particle changes over time. Differentiation allows us to convert the displacement function \( x = (t+1)^2 \) into a velocity function.

• When we talk about differentiating \( x \) with respect to \( t \), it means calculating \( \frac{dx}{dt} \), which represents the velocity of the particle.
• Velocity in physics is the rate of change of displacement with respect to time, which is exactly what differentiation facilitates.
• By differentiating \( x = (t+1)^2 \), we obtained \( v = 2(t+1) \), showing how the velocity directly relates to the change in time \( t \).

Differentiation thus helps us move from understanding displacement to capturing the more dynamic concept of velocity.

time-dependent displacement

Time-dependent displacement is a central concept when analyzing motion. It is essential because it describes how an object's position changes over time. In the given exercise, displacement is not static but changes as time changes, specified by \( \sqrt{x} = t + 1 \).
By rearranging this equation, we find how displacement \( x \) varies as \( x = (t+1)^2 \). It vividly illustrates that the position \( x \) grows as \( t \) increases, showing a quadratic dependence which implies accelerating motion.

• This relationship means displacement isn't constant but varies, signifying motion along the x-axis.
• Analyzing how \( x \) changes over time gives insight into the particle's velocity and how it accelerates.
• Understanding time-dependence helps predict future positions and evaluate motion characteristics.

This concept is pivotal in physics as it lays the groundwork for more complex analyses involving velocity and acceleration.