Question

A balloon is at a height of 81m and is ascending upwards with a velocity of 12m/s. a body of 2kg is dropped from it.the body will reach the surface in

Answer 1

$\Huge{\underline{\underline{\mathfrak{Correct \ Question \colon}}}}$

A balloon at a height of 81m from the ground is ascending upwards with a velocity 12 m/s,if a stone of mass 2 Kg is dropped from the ascending ballon. Find the time taken by the stone to reach the ground surface

$\Huge{\underline{\underline{\mathfrak{Answer \colon}}}}$

The initial velocity of the stone is equal to the velocity of the ascending ballon

• Initial Velocity,u = 12 m/s

• Mass of the stone,m = 2 Kg

The ballon is at 81 m above the ground

• Distance,s = 81 m

Using the Kinematic Equation,

$\huge{ \boxed{ \boxed{ \tt{s = ut + \frac{1}{2}at {}^{2} }}}}$

Stone is undergoing free fall.

• Acceleration,a = 10 m/s² [According to Sign Convention]

To find

Time taken by the stone to reach the ground

Substituting the values,we get :

$\sf{81 = 12t + \frac{1}{2}.10.t {}^{2} } \\ \\ \leadsto \: \sf{81 = 12t \: + 5t {}^{2} } \\ \\ \large{\leadsto \: \sf{5t {}^{2} + 12t - 81 = 0 }}$

Now,

On comparing with ax² + bx + c = 0,

a = 5,b = 12 and c = - 81

Discriminant,d = b² - 4ac

$\implies \: \sf{d = (12) {}^{2} - 4(5)( - 81) } \\ \ \\ \implies \: \sf{d = 144 + 1620} \\ \\ \large{ \sf{ \implies \: d = 1764}}$

Applying the Quadratic Formula,

$\huge{ \boxed{ \boxed{ \sf{t = \frac{ - b \pm \: \sqrt{d} }{2a} }}}}$

Putting the values,we get :

$\rightarrow \: \sf{t = \frac{ - 12 \pm \sqrt{1764} }{2(5)} } \\ \\ \rightarrow \: \sf{t = - \frac{ - 12 \pm \: 42}{10} } \\ \\ \rightarrow \: \sf{t = \frac{ - 12 - 42}{10} \: or \: \: \frac{ - 12 + 42}{10} } \\ \\ \rightarrow \: \sf{t = \frac{ - 54}{10} \: \: or \: \: \frac{30}{10} } \\ \\ \huge{ \rightarrow \: \underline{ \boxed {\green{ \sf{t = 3s}}}}}$

$\sf{\therefore \ the \ time \ taken \ by \ the \ stone \ to \ reach \ the \ ground \ is \ 3s }$