Differentating v(s) with respect to time (d v)/(d t)=(a)/(2 √(s)) (d s)/(d t)=(a)/(2 √(s)) a √(s

step 1 It is given that velocity of the locomotive varies as v equal to a under root as where as is dis

Differentating v(s) with respect to time (d v)/(d t)=(a)/(2...

Differentating $v(s)$ with respect to time $$ \frac{d v}{d t}=\frac{a}{2 \sqrt{s}} \frac{d s}{d t}=\frac{a}{2 \sqrt{s}} a \sqrt{s}=\frac{a^{2}}{2}=w $$ (As locomotive is in unidrectional motion) Hence force acting on the locomotive $F=m w=\frac{m a^{2}}{2}$ Let, at $v=0$ at $t=0$ then the distance covered during the first $t$ seconds $$ s=\frac{1}{2} w t^{2}=\frac{1}{2} \frac{a^{2}}{2} t^{2}=\frac{a^{2}}{4} t^{2} $$ Hence the sought work, $A=F s=\frac{m a^{2}}{2} \frac{\left(a^{2} t^{2}\right)}{4}=\frac{m a^{4} t^{2}}{8}$

Differentating $v(s)$ with respect to time
$$
\frac{d v}{d t}=\frac{a}{2 \sqrt{s}} \frac{d s}{d t}=\frac{a}{2 \sqrt{s}} a \sqrt{s}=\frac{a^{2}}{2}=w
$$
(As locomotive is in unidrectional motion) Hence force acting on the locomotive $F=m w=\frac{m a^{2}}{2}$ Let, at $v=0$ at $t=0$ then the distance covered during the first $t$ seconds
$$
s=\frac{1}{2} w t^{2}=\frac{1}{2} \frac{a^{2}}{2} t^{2}=\frac{a^{2}}{4} t^{2}
$$
Hence the sought work, $A=F s=\frac{m a^{2}}{2} \frac{\left(a^{2} t^{2}\right)}{4}=\frac{m a^{4} t^{2}}{8}$

Key Concepts

Work-Energy Principle

The work-energy principle connects the work done by forces to the change in kinetic energy of an object. It asserts that work, defined as the product of force and displacement, can alter the energy state of a system, making it a vital concept in both mechanics and energy analysis.

Newton's Second Law

Newton's second law states that the force acting on an object is equal to the mass of the object multiplied by its acceleration. This law is a key principle in mechanics, linking dynamic quantities like acceleration to the forces that produce them.

Kinematic Equations

Kinematic equations describe the motion of objects by relating displacement, velocity, acceleration, and time. These equations are particularly useful in cases of constant acceleration, providing formulas that connect these quantities and support the analysis of motion in physics.

Differentiation with Respect to Time

Differentiation with respect to time involves computing the derivative of a function where the independent variable is time. This is essential in dynamics as it allows us to determine quantities like velocity and acceleration from position functions, even when the position is given as a function of another variable.

Chain Rule

The chain rule is a fundamental principle in calculus used to differentiate composite functions. When a variable is a function of another variable, which in turn is a function of time, the chain rule allows us to relate the rate of change of the outer function with respect to time to the rate of change of the inner function.

Please give Ace some feedback