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A balloon is ascending at 12.4 m/s at a height of 81.3 m above the ground when a package is dropped. (a) With what speed does the package hit the ground? (b) How long did it take to reach the ground?

A balloon is ascending at 12.4 m/s at a height of 81.3 m above the ground when a package is dropped. (a) With what speed does the package hit the ground? (b) How long did it take to reach the ground?

Grade:upto college level

1 Answers

Navjyot Kalra
askIITians Faculty 654 Points
6 years ago
Given:
232-968_2.PNG
It is important to note that when the package will be thrown from the balloon, the initial speed of the package will be equal to the initial speed of the balloon. From the reference of the observer at ground, the package will have an initial velocity vector in upward direction, the package will appear to transcend upward and then fall back on Earth.
However the origin or reference for the motion of motion is at the balloon itself.
If we ignore the effect of the air resistance, the package will slow down under the action of acceleration due to gravity and after attaining some height, it will begin to fall on earth, thereby accelerating downwards.
Therefore it is better to divide the problem into the upward motion and the downward motion.
When the package will attain the maximum height (say h), its final speed will be zero. Therefore one can calculate the maximum height using the equation of kinematics as:
232-1517_3.PNG
Where v is the final speed of the package,v0 is the magnitude of initial speed, and x is the maximum height relative to the position when the package was dropped.
The acceleration acting on the package is -g because the Earth gravitational force slows it down by decelerating at rate equal to g .
Substitute the given values in the equation above to have
232-2308_4.PNG
Therefore the package rise up to 7.84.
232-164_5.PNG
232-1043_6.PNG
Therefore the final speed of the package, with which it hits the ground is 41.8 m/s .
(b) Let us assume that the time taken by the package to reach the ground from its maximum height is t1 . Therefore one can calculate the time using the equation of kinematics as:
232-1216_7.PNG
In the above equation, v is the final speed of the package when it hits the ground, is its initial speed and a is the acceleration acting on the package and is equal to g.
Substitute the values in the equation above to have
In the above equation, is the final speed of the package when it reaches the maximum height which is zero, is its initial speed of the package when it was dropped, and is the acceleration acting on the package and is equal to
The negative sign of the acceleration accounts for the fact that the gravity decelerates the package when it moves upwards. Also the initial speed of the package is equal to the initial speed of the balloon that is .232-1389_8.PNG
In the above equation, v is the final speed of the package when it reaches the maximum height which is zero, v0 is its initial speed of the package when it was dropped, and a is the acceleration acting on the package and is equal to -g.
The negative sign of the acceleration accounts for the fact that the gravity decelerates the package when it moves upwards. Also the initial speed of the package is equal to the initial speed of the balloon that is 12.4m/s .
Substitute the corresponding values in the equation above to have
232-2321_9.PNG
232-849_10.PNG
Therefore the package reaches the ground after 5.25 s.

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