A balloon is at a height of 81 m and is ascending upwards with a velocity of 12m /s. A body of 2 kg weight is dropped from it. If g= 10m/s², the body will reach the surface of the earth in

To find the time it takes for the dropped body to reach the ground, we can analyze its motion using the equations of kinematics under constant gravity.

1. Understanding the Initial Conditions:
When a body is dropped from an ascending balloon, it inherits the velocity of the balloon at that exact moment. Therefore, the body's initial velocity is directed upwards.
Let us choose the upward direction as positive and the downward direction as negative:
- Initial velocity of the body ($u$) = +12 m/s (upward)
- Acceleration due to gravity ($a$) = -10 m/s² (downward)
- Vertical displacement ($s$) when the body reaches the ground = -81 m (downward from the point of release)

2. Applying the Equation of Motion:
We use the second equation of motion to relate displacement, initial velocity, acceleration, and time ($t$):

$s=ut+\frac{1}{2}a{t}^2$

Substituting the values into the equation:

$-81=12t+\frac{1}{2}(-10)t^2$

$-81=12t-5t^2$

Rearranging this into the standard quadratic equation form ($A{t}^2+Bt+C=0$):

$5t^2-12t-81=0$

3. Solving the Quadratic Equation:
We can use the quadratic formula to solve for $t$:

$t=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Here, $a=5$, $b=-12$, and $c=-81$. Let us calculate the discriminant first:

$b^2-4ac={(-12)}^2-4\left(5\right)(-81)$

$b^2-4ac=144+1620=1764$

Now, finding the square root of 1764:

$\sqrt{1764}=42$

Substitute this back into the formula for $t$:

$t=\frac{12\pm 42}{10}$

This gives two possible values for $t$:

$t_1=\frac{12+42}{10}=\frac{54}{10}=5.4\text{s}$
$t_2=\frac{12-42}{10}=-3\text{s}$

Since time cannot be negative, we reject the negative root. Thus, the body will reach the surface of the earth in 5.4 seconds.