A hot air balloon along with its basket and contents has mass M. It is descending at a constant acceleration a less than the acceleration due to gravity. When the content of mass m is released from the hot air balloon, it starts rising upwards with the acceleration of same magnitude a. What is the value of m?
(A) \frac{\alpha}{M(a+g)}
(B) \frac{2\alpha}{M(a+g)}
(C) \frac{\alpha+g}{M\alpha}
(D) \frac{2\alpha}{M(a+g)}
Answer (2)
To find the mass m released from the hot air balloon, we applied Newton's Second Law, resulting in m = g 2 M a . This solution indicates an error in the provided answer choices, as none match precisely. A re-evaluation of the options may be warranted based on this calculation.
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Let's solve this problem step-by-step.
To understand the problem, we must use Newton's 2nd Law, which states:
F = ma
Here, we deal with the forces acting on a hot air balloon.
Initial Conditions :
The balloon's total mass is M .
The balloon is descending with an acceleration a (where a < g , g being the acceleration due to gravity).
Weight of the balloon is M g and the net force acting on it while descending is M a .
Force Balance While Descending :
The net force equation while descending can be given as:
M g − F u p = M a
where F u p is the upward buoyant force.
Solving, we have:
F u p = M g − M a = M ( g − a )
After Ejecting Mass m :
Now when mass m is released, the balloon's mass becomes M − m , and it starts to rise with the same magnitude of acceleration a .
The force equation when the balloon starts rising:
F u p − ( M − m ) g = ( M − m ) a
Substituting for the upward force F u p from earlier:
M ( g − a ) − ( M − m ) g = ( M − m ) a
Expanding and rearranging gives:
M g − M a − M g + m g = M a − ma
Combining like terms:
m g = 2 M a
Solving for m , we get:
m = g 2 M a
Therefore, substituting m expression into the given options:
Answer Choice : The correct answer is not directly listed among the choices. There might be an error in the provided options or in the interpretation. However, based on derivation and typical physics question constructs, the correct format should reflect a proportional relationship in terms of M , a , and g .
Consequently, the explanation aligns closely with an adjusted application of one such option, focusing primarily on prior calibration between constants.
