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A body A is given an initial velocity v in the X -direction...

A body $\mathrm{A}$ is given an initial velocity $\mathrm{v}$ in the $\mathrm{X}$ -direction at $\mathrm{t}=0$; its velocity is increased by $\mathrm{v}$ in the same direction at the end of every second. Another body B starting from zero velocity is given a uniform acceleration 'a' in X-direction. Both travel the same distance for $\mathrm{n}$ second. (a) Acceleration $\mathrm{a}=\mathrm{v}\left(1+\frac{1}{\mathrm{n}}\right)$. (b) The distance travelled by the first body in $20 \mathrm{~s}$ is $210 \mathrm{v}$ metre. (c) If $\mathrm{n}$ is large, in 5 th second both travel the same distance. (d) If $n$ is large, for even values of $n$, in the $\left(\frac{n}{2}\right)$ second both travel equal distance; if $n$ is odd, at $\frac{n+1}{2}$ second they travel equal distance.

A body $\mathrm{A}$ is given an initial velocity $\mathrm{v}$ in the $\mathrm{X}$ -direction at $\mathrm{t}=0$; its velocity is increased by $\mathrm{v}$ in the same direction at the end of every second. Another body B starting from zero velocity is given a uniform acceleration 'a' in X-direction. Both travel the same distance for $\mathrm{n}$ second.
(a) Acceleration $\mathrm{a}=\mathrm{v}\left(1+\frac{1}{\mathrm{n}}\right)$.
(b) The distance travelled by the first body in $20 \mathrm{~s}$ is $210 \mathrm{v}$ metre.
(c) If $\mathrm{n}$ is large, in 5 th second both travel the same distance.
(d) If $n$ is large, for even values of $n$, in the $\left(\frac{n}{2}\right)$ second both travel equal distance; if $n$ is odd, at $\frac{n+1}{2}$ second they travel equal distance.

Key Concepts

Uniformly Accelerated Motion

This concept deals with motion under a constant acceleration where the velocity increases linearly over time and the displacement follows a quadratic relationship with time. It is characterized by equations like v = u + at and s = ut + 1/2 at², and is fundamental when analyzing objects in free-fall or any system under constant acceleration.

Stepwise or Discrete Acceleration

This concept involves scenarios where the acceleration is not applied continuously but in increments at distinct time intervals. In such cases, the object’s speed increases in discrete steps rather than smoothly, requiring the use of summation techniques to determine the overall displacement and velocity over a period.

Arithmetic Series in Motion Analysis

When velocity or displacement increments are added at regular intervals, they form an arithmetic series. Summing these increments over time is essential to calculate total distance or velocity in systems with discrete changes. This approach bridges the gap between stepwise motion and its continuous counterpart by using series summation formulas.

Comparative Motion Analysis

This involves comparing the outcomes of different motion models—such as discrete versus continuous acceleration—to determine conditions under which certain parameters (like distance travelled) are equivalent. This analysis is critical when matching trajectories or displacements in systems following different kinematic rules.

Limit Analysis in Kinematics

This concept studies the behavior of kinematic quantities as certain parameters, such as the number of time intervals, become very large. Limit analysis helps in understanding the convergence or divergence between different models of motion, providing insights into the system’s behavior under extreme conditions or in asymptotic scenarios.

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