Velocity of a bullet changes from u to v after passing...

Velocity of a bullet changes from $u$ to $v$ after passing through a board of thickness $d$. Force of resistance is directly proportional to the velocity. Time of motion of bullet in the board is given by (a) $\frac{d(u-v)}{u v \log _{\ell} \frac{u}{v}}$ (b) $\frac{d u}{v \log _e \frac{u}{v}}$ (c) $\frac{d v}{u \log _e \frac{u}{v}}$ (d) $\frac{d(v-u)}{u v \log _{\ell} \frac{v}{u}}$

Velocity of a bullet changes from $u$ to $v$ after passing through a board of thickness $d$. Force of resistance is directly proportional to the velocity. Time of motion of bullet in the board is given by
(a) $\frac{d(u-v)}{u v \log _{\ell} \frac{u}{v}}$
(b) $\frac{d u}{v \log _e \frac{u}{v}}$
(c) $\frac{d v}{u \log _e \frac{u}{v}}$
(d) $\frac{d(v-u)}{u v \log _{\ell} \frac{v}{u}}$

Key Concepts

Integration Techniques

Integration is a core mathematical tool used to solve differential equations derived from physical laws. When handling problems involving variable forces, integration allows the analyst to determine relationships between variables such as velocity, time, and displacement, often leading to solutions expressed in terms of definite integrals that capture the cumulative effect of the force over an interval.

Logarithmic Functions in Motion Analysis

Logarithmic functions often appear as a result of integrating equations where the force is proportional to velocity. These functions help in expressing the relationship between initial and final conditions of a motion scenario, encapsulating the exponential decay or growth processes that typically arise in systems with resistive forces.

Resistive Forces

Resistive forces are forces that oppose the motion of an object, often dependent on properties of the object’s speed. In many problems in mechanics, especially those involving drag or friction, the resistive force can be modelled as proportional to the velocity or its square, fundamentally altering the motion relative to idealized cases with no resistance.

Differential Equations in Dynamics

In dynamics, when forces depend on velocity, the resulting equations of motion are typically differential equations. These equations express the rate of change of an object’s velocity or position with respect to time and require techniques from calculus to solve them, thereby linking the physical behaviour of the system with mathematical models.

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