A gas balloon is powered by engines which can give it a constant but adjustable upward acceleration. At an instant of time when it is 509.5 m above the groundand has an upward velocity of 98.1 m/s the engines jam for 13 seconds.The engines restart after the period of 13 s . Now, to what constant acceleration should the engines be adjusted so as to reach the ground with zero velocity. Take the acceleration due to gravity as 9.81 m /s^{2} and assume that the gas balloon moves only in the vertical direction. a) 0.453 m / s^{2} . b) 10.263 m / s^{2} . c) 9.810 m / s^{2} . d) 11.810 m / s^{2} . 

Let the points of interest be marked as A, B, C and D as shown in the figure. Let s_{i}, v_{i}, a_{i} and t_{i} represent the displacement from the ground, instantaneous velocity, acceleration and time at the position i, where i = A, B, C and D. Let s_{ij} and t_{ij} represent the displacement and the time taken between the points i and j, where i, j = A, B, C and D and i not equal to j. At Point A ( Where the engines fail ): Engines fail and the acceleration of the gas balloon is only due to the earths gravity. s_{A}
= 509.5 m. v_{A}
= 98.1 m/s . At Point B ( At te highest point of the path ): Assuming that the acceleration till it reaches the highest point, B, is due to gravity alone; the time taken can be calculated from, v_{B}
= V_{A} + a t_{B}. Since 10 s is less than the time of jam, 13 s, our assumption above is justified. From, v_{B}^{2}
 V_{A}^{2} = 2 a s_{AB} s_{A} + s_{B} = 509.5 + 490.5 = 1000 m. At Point C ( Where the engines restart ) : The point C is the point at which the engines restart. This point is 3 seconds ( 13  10 ) seconds away from point B. The displacement s_{BC} and the velocity at C , v_{C}, are calculated as follows s_{BC}
= v_{B} t_{BC} + 0.5 a t_{BC}^{2}. The velocity at C
is given by Since it just touches the ground at D , the velocity v_{D} = 0 m/s.The required net acceleration, a_{net}, can be calculated from the following equations. v_{D}^{2}
 v_{C}^{2} = 2 (a_{net})(s_{C}) Acceleration due to
gravity is always present and is acting downwards at any instant of time.
