Question

A balloon starts rising from the ground from rest with on upward acceleration \[2m/{{s}^{2}}\]. Just after is a stone dropped from it. The time taken by stone to strike the ground is nearly.
A. \[0.3s\]
B. \[1.7s\]
C. 1s
D. \[1.4s\]

Answer 1

Hint: Since, a balloon starts rising from ground from rest with an upward acceleration \[2m/{{s}^{2}}\] after 1s, and a stone is dropped from it. So we have to use the formula \[v=u+at\] and \[{{v}^{2}}-{{u}^{2}}=2as\] to find the time taken by stone to strike the ground.

Complete answer:
As we know that when the stone is dropped from the balloon, its initial velocity is the same as the velocity of the balloon at that instant.
The upward acceleration is 2m/s². The balloon starts from rest, So \[u=0\].The balloon rises for 1 sec before the stone is dropped, hence we have
\[\begin{align}
 & v=u + at \\
 & v=0+2=2m/s \\
\end{align}\]
In the vertically upward direction. This is the
initial velocity of the stone after belonging dropped.
The distance moved up by the balloon in 1 second is
\[\begin{align}
  & {{v}^{2}}-{{u}^{2}}+2as \\
 & {{v}^{2}}=2as \\
 & s=\dfrac{{{v}^{2}}}{2as}=\dfrac{4}{2\times 2}=1m \\
\end{align}\]
Hence, the stone falls by 1 m before hitting the ground.
Now, the acceleration on the store after being
dropped is \[g=9.8m/{{s}^{2}}\]
From equation of motion,
\[\begin{align}
  & s=ut+\dfrac{1}{2}a{{t}^{2}} \\
 & \Rightarrow 1=-2t+\dfrac{1}{2}9.8\times {{t}^{2}} \\
 & \Rightarrow 4.9{{t}^{2}}-2t-1=0 \\
 & \therefore t=\dfrac{2\pm \sqrt{{{\left( -2 \right)}^{2}}-4\times 4.9\times \left( -1 \right)}}{2\times 9.8} \\
 & =\dfrac{2+\sqrt{4+19.6}}{2\times 9.8}=\dfrac{2+\sqrt{23.6}}{9.8} \\
\end{align}\]
Since time can't be the sol \[-ve\]. So, we use the value \[t=\dfrac{2+\sqrt{23.6}}{9.8}\]
\[t=0.7\sec \]
Hence, the stone will strike the ground \[0.75\] after being dropped from the balloon.

So, the correct answer is “Option B”.

Note:
Be careful; while using the formula of equation of motion \[v=u+at\] and \[{{v}^{2}}-{{u}^{2}}=2as\] to find the time taken by stone to strike the ground. Put the value exactly at the same time.