Correct option is C. reach the ground in 4s
Displacement of the balloon, s = ut + \(\frac 12\)at2
The velocity of the balloon, v = u + at
Where,
s = displacement (m)
u = initial velocity (m/s)
v =the final velocity (m/s)
a =the acceleration of the body (ms−2)
t = the time is taken (s)
We are Given with, a balloon starts rising from ground therefore initial velocity u=0 m/s
As it moves upwards its acceleration will be 1.25ms−2 and it reaches a certain height then a stone falls from the balloon after 8s.
i.e. u = 0 m/s,
a = 1.25ms−2
Now we will calculate the distance of the stone above the ground about which it begins to fall from the balloon.
Here, Let s = h, u = 0m/s, and a = 1.25ms−2, t = 8s
Then, substituting the values in the formula s = ut + \(\frac 12\)at2 we get,
h = 0 + \(\frac 12\)(1.25)82
Therefore, h = 40m.
Stone covers a distance of 40m before reaching the ground.
Next, we will find out the velocity of stone at height 40m,
The velocity of the balloon at this height can be obtained from the formula
v = u + at
Here, u = 0, a = 1.25ms−2, t = 8s
Then we get,
v = 0 + (1.25)8 = 10ms−1
This velocity becomes the initial velocity (u’) of the stone as the stone falls from the balloon from height h.
Therefore we have now, the initial velocity at height h, u’= 10ms-1
Now let us calculate the total time taken by the stone to reach the ground.
Total motion of the stone, is given by an equation,
h = \(\frac 12\)gt2 − u′t
Here, h = 40m, u’ = 10ms−1 and t is the time taken by the stone to reach the ground.
Substituting values in kinematic equation, we get
−40 = 10t − \(\frac 12\) × 10t2
After solving,
−40 = 10t − 5t2
5t2 − 10t − 40 = 0
Divide the above equation by 5, we get
t2 − 2t − 8 = 0
After factorization,
t2 − 4t + 2t − 8 = 0
t(t − 4) + 2(t − 4) = 0
Therefore t - 4 = 0 or t + 2 = 0
Then the value of t comes to be, t = 4 or -2
Ignoring the negative value of the time we get, t = 4s.