A man in a car at location $Q$ on a straight highway is moving with speed $v$. He decides to reach a point $P$ in a field at a distance d from the highway (point $M$) as shown in the figure. Speed of the car in the field is half to that on the highway. What should be the distance $RM$, so that the time taken to reach $P$ is minimum ?
Let the distance $QM =l$ and distance $RM =x$. Time to reach from $Q$ to $R$ is $t_{1}=\frac{l-x}{v}$ Time to reach from $R$ to $P$ is $t_{2}=\frac{\sqrt{x^{2}+d^{2}}}{v / 2}$ Therefore,$\,\,\,\, t=t_{1}+t_{2}=\frac{l-x}{v}+\frac{\sqrt{x^{2}+d^{2}}}{v / 2}$ On differentiating, we get $\frac{d t}{d x}=\frac{0-1}{v}+\frac{1}{v / 2} \frac{1}{2 \sqrt{x^{2}+d^{2}}} \times 2 x $ $\Rightarrow \frac{d t}{d x}=\frac{-1}{v}+\frac{2 x}{v \sqrt{x^{2}+d^{2}}}$
The motion in a straight lineis an object changes its position with respect to its surroundings with time, then it is called in motion. It is a change in the position of an object over time. It is nothing but linear motion.
Types of Linear Motion:
Linear motion is also known as the Rectilinear Motion which are of two types:
Uniform linear motion with constant velocity or zero acceleration: If a body travels in a straight line by covering an equal amount of distance in an equal interval of time then it is said to have uniform motion.
Non-Uniform linear motion with variable velocity or non-zero acceleration: Not like the uniform acceleration, the body is said to have a non-uniform motion when the velocity of a body changes by unequal amounts in equal intervals of time. The rate of change of its velocity changes at different points of time during its movement.
