Behavior of Water Stream Flowing Vertically from Hose

To determine the correctness of the statements, we analyze the fluid dynamics governing the motion of a stream of water:

1. Analysis of STATEMENT-2:
   According to the principle of conservation of mass, for a steady, streamline flow of an incompressible fluid, the mass flow rate through any cross-section of the flow tube is constant. Since the fluid's density $\rho$ is constant:
   $\frac{dm}{dt} = \rho A v = \text{constant}$
   Dividing by the constant density $\rho$, we obtain the equation of continuity for volume flow rate $Q$:
   $Q = A v = \text{constant}$
   where:

• $A$ is the cross-sectional area of the fluid stream.
• $v$ is the speed of the fluid at that cross-section.

Therefore, the volume flow rate remains constant throughout the steady flow of an incompressible fluid.
STATEMENT-2 is True.

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1. Analysis of STATEMENT-1:
   From the equation of continuity, the cross-sectional area of the stream is inversely proportional to its velocity:
   $A = \frac{Q}{v} \implies A \propto \frac{1}{v}$

• When the hose is held vertically upwards:
  As the water moves upward against the acceleration due to gravity ($g$), its velocity decreases ($v$ decreases with height). Since $A \propto \frac{1}{v}$, the cross-sectional area of the stream increases as it rises. As a result, the stream broadens and tends to spread out like a fountain.

• When the hose is held vertically downwards:
  As the water falls downward, gravity accelerates the water, causing its velocity to increase ($v$ increases with distance fallen). Consequently, the cross-sectional area $A$ must decrease, causing the stream to narrow down.

Therefore, STATEMENT-1 is True.

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1. Relationship between the statements:
   The widening of the stream when flowing upward and the narrowing of the stream when flowing downward are direct consequences of the constancy of the volume flow rate ($A v = \text{constant}$). Hence, STATEMENT-2 is the correct explanation for STATEMENT-1.

Correct Answer: (A)