Question:
A hot air balloon is descending with a constant acceleration of \(2 \, \text{m/s}^2\). The mass of the balloon and its contents is 600 kg. Assuming the volume of the balloon is kept constant so that the buoyant force remains the same, the mass of the balloon should be changed so that it starts ascending with the same acceleration. (Take \(g = 10 \, \text{m/s}^2\)).
A hot air balloon is descending with a constant acceleration of \(2 \, \text{m/s}^2\). The mass of the balloon and its contents is 600 kg. Assuming the volume of the balloon is kept constant so that the buoyant force remains the same, the mass of the balloon should be changed so that it starts ascending with the same acceleration. (Take \(g = 10 \, \text{m/s}^2\)).
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In buoyancy problems, always remember: if volume is constant, buoyant force remains constant. Then use Newton’s second law separately for each motion condition.
Updated On: Jun 20, 2026
- 400 kg
- 200 kg
- 100 kg
- 500 kg
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The Correct Option is A
Solution and Explanation
Step 1: Understand the physical situation.
The balloon is initially descending with acceleration \(a = 2 \, \text{m/s}^2\). The mass of system is \(m = 600 \, \text{kg}\). The buoyant force \(B\) is constant because volume is constant. Forces acting are: \[ \text{Weight} = mg \quad \text{(downward)}, \quad \text{Buoyant force} = B \quad \text{(upward)} \] We take downward direction as positive for initial motion.
Step 2: Apply Newton’s second law for descending motion.
For downward acceleration: \[ mg - B = ma \] Substitute values: \[ 600 \times 10 - B = 600 \times 2 \] \[ 6000 - B = 1200 \] \[ B = 6000 - 1200 = 4800 \, \text{N} \] So, the buoyant force is: \[ B = 4800 \, \text{N} \]
Step 3: Condition for upward motion with same acceleration.
Now the balloon is required to ascend with acceleration \(a = 2 \, \text{m/s}^2\). Let the new mass be \(m\). Now forces act such that upward acceleration is positive: \[ B - mg = ma \]
Step 4: Substitute known values.
\[ 4800 - 10m = 2m \]
Step 5: Solve the equation step by step.
Bring like terms together: \[ 4800 = 10m + 2m \] \[ 4800 = 12m \] Now divide both sides: \[ m = \frac{4800}{12} \] \[ m = 400 \, \text{kg} \]
Step 6: Final interpretation.
The mass of the balloon must be reduced to \(400 \, \text{kg}\) so that the same buoyant force can produce an upward acceleration of \(2 \, \text{m/s}^2\). This ensures the net upward force is sufficient to overcome weight and produce required acceleration.
\[ \boxed{400 \, \text{kg}} \]
The balloon is initially descending with acceleration \(a = 2 \, \text{m/s}^2\). The mass of system is \(m = 600 \, \text{kg}\). The buoyant force \(B\) is constant because volume is constant. Forces acting are: \[ \text{Weight} = mg \quad \text{(downward)}, \quad \text{Buoyant force} = B \quad \text{(upward)} \] We take downward direction as positive for initial motion.
Step 2: Apply Newton’s second law for descending motion.
For downward acceleration: \[ mg - B = ma \] Substitute values: \[ 600 \times 10 - B = 600 \times 2 \] \[ 6000 - B = 1200 \] \[ B = 6000 - 1200 = 4800 \, \text{N} \] So, the buoyant force is: \[ B = 4800 \, \text{N} \]
Step 3: Condition for upward motion with same acceleration.
Now the balloon is required to ascend with acceleration \(a = 2 \, \text{m/s}^2\). Let the new mass be \(m\). Now forces act such that upward acceleration is positive: \[ B - mg = ma \]
Step 4: Substitute known values.
\[ 4800 - 10m = 2m \]
Step 5: Solve the equation step by step.
Bring like terms together: \[ 4800 = 10m + 2m \] \[ 4800 = 12m \] Now divide both sides: \[ m = \frac{4800}{12} \] \[ m = 400 \, \text{kg} \]
Step 6: Final interpretation.
The mass of the balloon must be reduced to \(400 \, \text{kg}\) so that the same buoyant force can produce an upward acceleration of \(2 \, \text{m/s}^2\). This ensures the net upward force is sufficient to overcome weight and produce required acceleration.
\[ \boxed{400 \, \text{kg}} \]
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