Question

The following graph shows the variation of velocity of a rocket with time. Then the maximum height attained by the rocket is-

(A) $1.1\,Km$
(B) $5\, Km$
(C) $55\,Km$
(D) None of these

Answer 1

Analyze the given graph of the velocity versus time. The area covered by the velocity-time graph provides the answer for the maximum height traveled. Use the formula of the area of the triangle given below, apply it to the given graph to find the area covered by the graph.
Formula used:
The area of the triangle is given by
$A = \dfrac{1}{2}bh$
Where $A$ is the area of the triangle, $b$ is the base and $h$ is the height of the triangle.

Complete step-by-step solution:
The given graph in the question represents the velocity of the rocket with respect to time. In order to find the maximum height at which the rocket travels, the integral part of the velocity with respect to time is taken.
${h_{\max }} = \int {vdt}$
The maximum height is calculated by the area covered under the line $vt$.
${h_{\max }} = A$
Substituting the area of the triangle in the above step, we get
${h_{\max }} = \dfrac{1}{2}bh$
Substitute the value of the base of the triangle and the height of the triangle from the graph of the velocity versus time given.
${h_{\max }} = \dfrac{1}{2} \times 110 \times 1000$
By performing a various arithmetic operation in the above step, we get
${h_{\max }} = 55\,Km$
Hence the maximum height at which the rocket moves is obtained as $55\, Km$.
Thus the option (C) is correct.

Note: Don’t be confused that the maximum height of the rocket is $1000$ . This is wrong, it is only applicable when the graph is displacement versus time. Since it is velocity versus time, the highest value $1000$ is the maximum velocity value, so it cannot be taken as the maximum height. It is obtained by taking the area covered by the graph.