Question

Superman is going on a vacation to his home planet krypton from earth via. Sun. (Let's assume for a moment that Krypton is not destroyed). His energy v/s distance travelled graph is as shown below:

Find the number of positions where $\frac{dE}{ds}$=0. Based on this find out the number of stars that Super man crossed on his way home from earth. (I hope you know that superman's source of energy is sunlight or star light so the more closer he goes to a star the more energy he has)

• A. number of positions : 3, number of stars : 3
• B. number of positions : 2, number of stars : 2
• C. number of positions : 3, number of stars : 2
• D. number of positions : 2, number of stars : 3

Answer 1

The correct option is C

number of positions : 3, number of stars : 2

So there are 3 points where the tangent is parallel to the axis.
Now before 150 million km mark his energy is quite low relatively but then it keeps increasing and becomes maximum at 150 mark then starts decreasing. This means at that place there would have been a star. At 300 million km superman's energy is the lowest, so there cannot be a star there, whereas 1000 million km his energy is again maximum. So we can conclude there is another star.
Conclusion: $\frac{dy}{dx}=0$ (maxima or minima exits)

You will also notice tan $\theta$ to be positive before 150 million km but negative afterwards. $△tan\theta$ is negative i.e., rate of change of $\left(tan\theta \right)$ is negative or rate of change of energy is negative
$\Rightarrow \frac{d^{2}E}{d{s}^2}=negative.$
Also before 300 mark tan $\theta$ is negative while just after becomes positive
$\Rightarrow △tan\theta$ is positive
$\Rightarrow \frac{d^{2}E}{d{s}^2}=positive$
Conclusion 2:
To find maxima
$\frac{dy}{dx}=0;\frac{d^{2}y}{d{x}^2}=negative$
Eg. At 150 million km