Question

A balloon carrying some sand of mass \( M \) is moving down with a constant acceleration \( a_0 \). The mass \( m \) of sand to be removed so that the balloon moves up with double the acceleration \( a_0 \) is:

• A. m = \( \frac{2 M a_0}{a_0 + g} \)
• B. m = \( \frac{2 M a_0}{a_0 - g} \)
• C. m = \( \frac{3 M a_0}{g + 2 a_0} \)
• D. m = \( \frac{3 M a_0}{g - 2 a_0} \)

Hint:

In problems involving forces and accelerations, use Newton's second law to relate the forces and accelerations, and then solve for the unknown quantities.

Answer 1

The Correct Option is C

We are given that the balloon moves downward with a constant acceleration \( a_0 \). Let the mass of sand to be removed be \( m \), and we want the balloon to move upward with a double acceleration \( 2a_0 \). 

Step 1: For downward motion, the net force acting on the balloon is: \[ F_{\text{down}} = M \cdot g - M \cdot a_0. \] 

Step 2: For upward motion, the net force acting on the balloon after removing mass \( m \) is: \[ F_{\text{up}} = (M - m) \cdot g + (M - m) \cdot 2a_0. \] 

Step 3: By equating the two forces, we can solve for \( m \), yielding the mass of sand to be removed: \[ m = \frac{3 M a_0}{g + 2 a_0}. \] 

Thus, the mass \( m = \frac{3 M a_0}{g + 2 a_0} \).