Problem 63 - Master Resource Book in JEE Main Physics

Question

A particle covers \(4 \mathrm{~m}, 5 \mathrm{~m}, 6 \mathrm{~m}\) and \(7 \mathrm{~m}\) in \(3 \mathrm{rd}, 4\) th, 5th and 6th second respectively. The particle starts (a) with an initial non-zero velcoity and moves with uniform acceleration (b) from rest and moves with uniform velocity (c) with an initial velocity and moves with uniform velcoity (d) from rest and moves with uniform acceleration

Step-by-Step Solution

Answer

(d) The particle starts from rest and moves with uniform acceleration.

Step 1: Understanding the Motion

The particle covers different distances ( 4 m, 5 m, 6 m, and 7 m) during successive seconds (3rd, 4th, 5th, and 6th). This suggests a change in velocity over time, implying acceleration.

Step 2: Determine Type of Motion

The fact that the distances covered by the particle increase by 1 meter each second supports a scenario with consistent acceleration rather than constant velocity.

Step 3: Apply Equation of Motion

Using the equation for distance in uniformly accelerated motion for the nth second: \[ S_n = u + \frac{a}{2}(2n-1) \]Check if our distances meet this formula with a constant acceleration (a) and initial velocity (u).

Step 4: Calculate Differences

Calculate the difference in distances covered in successive seconds, i.e., 5m - 4m = 1m, 6m - 5m = 1m, 7m - 6m = 1m. The constant increase indicates a uniform acceleration.

Step 5: Conclusion

The particle moved with a uniform acceleration starting from an initial velocity of zero, fitting the characteristics of uniform acceleration.

Key Concepts

Equation of motion

The equation of motion is a mathematical relationship used to describe the behavior of moving objects. It particularly applies when dealing with uniform acceleration, where the rate of change of velocity is constant. In our context, the equation for the distance covered by an object in the nth second is given by:\[ S_n = u + \frac{a}{2}(2n-1) \]Here, \( S_n \) is the displacement in the nth second, \( u \) is the initial velocity, and \( a \) is the constant acceleration. This equation helps us understand how both the initial velocity and the continuous application of acceleration contribute to the object's displacement over time. In our example, the particle increases its displacement by an equidistant measure over subsequent seconds, indicating a uniform acceleration is precisely at work.

Initial velocity

Initial velocity is the speed at which an object begins its motion. It is a critical component in determining how an object will move when subjected to external forces like acceleration. In many cases, including uniform acceleration scenarios, initial velocity is considered as time zero motion.

• If initial velocity \( u = 0 \), it means the object started from a standstill.
• If \( u eq 0 \), the object was already in motion.

The importance of initial velocity is seen in our scenario where it's integral to the constants in the motion equations. Without an initial velocity, an object won't follow the expected trajectory when only uniform acceleration is applied. For clarity, imagine pushing an object along a surface. If it already has an initial velocity, the way it accelerates due to your push will add to its existing motion, not just start a new one from zero.

Displacement in successive time intervals

Displacement refers to the change in position of an object over a period of time. In a uniform acceleration context, this is especially important as it helps quantify how far an object travels in each second or nth interval. Observing displacement over successive time intervals can reveal a lot about the object's motion type:

• Constant displacement suggests uniform velocity.
• Varying displacement, specifically increasing displacement, signifies uniform acceleration.

In the provided exercise, the particle shows increased displacement by 1 meter each second for the given intervals. This steady increment indicates that the particle is under the effect of uniform acceleration. By noting this pattern, it becomes evident that the subsequent amount it travels in each new interval isn't random but heavily influenced by consistent force application, supporting the conclusion of uniform acceleration beginning from an initial velocity.