Question

A stone is dropped from a balloon going up with a uniform velocity of \[10m/s\]. If the height of the balloon was \[56.25m\] when the stone was dropped, then the time after which the stone will strike the ground after it is dropped is
A.\[3s\]
B.\[3.5s\]
C.\[4s\]
D.\[4.5s\]

Answer 1

Hint: Here balloon is going up and at the same time stone is falling down from a certain height so velocity of balloon becomes equal to velocity of stone at the time of falling only direction is taken in the opposite order. So the velocity of the balloon is taken as positive and velocity of stone is negative.

Complete answer:
When balloon going up with velocity of\[10m/s\] reaches upto the height of \[56.25m\] then a stone is dropped from that balloon then velocity of stone will become \[-10m/s\](-ve sign represents that direction of falling stone is opposite to the direction of balloon)
Let us assume the velocity of stone is \[u=-10m/s\]
Height from ground \[h=56.25m\]
Stone is falling under gravity then acceleration \[g=10m/{{s}^{2}}\]
Now apply second equation of motion,
\[h=ut+\dfrac{1}{2}g{{t}^{2}}\]
\[56.25=-10t+\dfrac{1}{2}10{{t}^{2}}\]
\[56.25+10t-5{{t}^{2}}=0\]
\[5{{t}^{2}}-10t-56.25=0\]
On Solving we get the value of t,
\[t=\dfrac{10\pm \sqrt{100-4(5)(-56.25)}}{2(5)}\]
\[t=\dfrac{10\pm \sqrt{1225}}{10}\]
\[t=\dfrac{10\pm 35}{10}\]
Taking \[t\] as positive we get
\[t=4.5\operatorname{s}\]
Taking \[t\] as negative we get
\[t=-2.5s\](which is not possible)
So time taken by stone to reach the ground will be \[t=4.5\operatorname{s}\]

So option D is the correct answer.

Additional Information:
Equations of motion are used when the body is falling under gravity and acceleration becomes acceleration due to gravity. Its value becomes negative when a body is going away from earth and its value is taken as positive when the body is falling towards earth.

Note:
The main step is to remember that velocity of stone should be taken as negative because velocity is vector quantity if direction changes then minus sign should be implemented here balloon going up is treated as positive so stone falling down should be treated as negative.