Question

A particle travels according to the equation $y = 4x - \dfrac{{{x^2}}}{2}$ . Find the maximum height it achieves. Where $y$ is displacement in vertical direction and $x$ is displacement in horizontal direction.
A. $8\,m$
B. $4\,m$
C. $2\,m$
D. $1\,m$

Answer 1

In this problem the equation for displacement is given and the velocity will be calculated by differentiating the given equation and the value for $x$ will be calculated by taking velocity as zero at maximum height and on substituting the value of $x$ in the given equation the maximum height can be obtained.

Complete step by step answer:
Given: The displacement of $y$ in vertical direction in terms of $x$ in horizontal direction is
$y = 4x - \dfrac{{{x^2}}}{2}$ ……… $\left( 1 \right)$
Differentiating the above equation with respect to x,
$\dfrac{{dy}}{{dx}} = 4 - \dfrac{{2x}}{2}$ ……… $\left( 2 \right)$
For maximum height, velocity will be zero
Therefore, $\dfrac{{dy}}{{dx}} = 0$ ……… $\left( 3 \right)$ (velocity = $\dfrac{{dy}}{{dt}}$ )
Substituting equation $\left( 3 \right)$ in equation $\left( 2 \right)$
$0 = 4 - \dfrac{{2x}}{2}$
On simplifying the above equation,
$4 = \dfrac{{2x}}{2}$
Therefore, $x = 4$
Substituting the value of $x$ in equation $\left( 1 \right)$ , we get
${y_{\left( {x = 4} \right)}} = \left( {4 \times 4} \right) - \dfrac{{{4^2}}}{2}$
On simplifying the above equation, we get
$y = 16 - \dfrac{{16}}{2}$
$\therefore y = 16 - 8$
Therefore the maximum height is $\left( y \right) = 8m$ .

Hence, the correct option is A.

Additional information: Velocity is defined as the rate of change of displacement and it is a vector quantity which has both magnitude and direction. Equation for velocity is given as,
$V = \dfrac{{dy}}{{dt}}$ …….. $\left( 4 \right)$
And also velocity will be zero at maximum height because the kinetic energy will be zero at maximum height and kinetic energy is defined as the virtue of motion it depends on the velocity, Therefore the above equation can be re written as
$\dfrac{{dy}}{{dt}} = 0$ ………. $\left( 5 \right)$

Note: It should be noted that the velocity will be zero at maximum height since the kinetic energy will be zero at maximum height and potential energy will be maximum at maximum height. At minimum height, kinetic energy will be maximum at minimum height and potential energy will be zero at minimum height.