Distance of Efflux Jet in an Accelerating Lift System

To determine the distance $d$ where the water jet lands on the floor of the lift, we consider the motion in the non-inertial frame of reference of the lift.

1. General Formula for Efflux Jet Distance

Let:

• $H$ be the height of the water column above the hole.
• $h$ be the vertical distance from the hole to the floor of the lift.
• $g_{\text{eff}}$ be the effective acceleration due to gravity inside the lift's reference frame.

Speed of Efflux: Applying Bernoulli's principle in the reference frame of the lift, the velocity of the water jet coming out horizontally from the hole is given by Torricelli's Law: $v = \sqrt{2 g_{\text{eff}} H}$

Time of Flight: The vertical trajectory of the water jet inside the lift is governed by $g_{\text{eff}}$. The time $t$ required for the water stream to fall a vertical distance $h$ to reach the floor is: $h = \frac{1}{2} g_{\text{eff}} t^2 \implies t = \sqrt{\frac{2h}{g_{\text{eff}}}}$

Horizontal Distance ($d$): The horizontal range $d$ of the jet on the floor of the lift is: $d = v \times t = \sqrt{2 g_{\text{eff}} H} \times \sqrt{\frac{2h}{g_{\text{eff}}}} = 2\sqrt{H h}$

Notice that the effective acceleration due to gravity $g_{\text{eff}}$ cancels out completely. Therefore, as long as $g_{\text{eff}} > 0$, the distance $d$ is independent of the acceleration of the lift and remains constant at $d = 1.2\text{ m}$.

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2. Analysis of Individual Cases

• P. Lift is accelerating vertically up: The effective gravity is $g_{\text{eff}} = g + a > 0$. Since $d$ is independent of $g_{\text{eff}}$, $d = 1.2\text{ m}$. P matches with 1.

• Q. Lift is accelerating vertically down with $a < g$: The effective gravity is $g_{\text{eff}} = g - a > 0$. Since $d$ is independent of $g_{\text{eff}}$, $d = 1.2\text{ m}$. Q matches with 1.

• R. Lift is moving vertically up with constant speed ($a = 0$): The effective gravity is $g_{\text{eff}} = g$. Therefore, $d = 1.2\text{ m}$. R matches with 1.

• S. Lift is falling freely ($a = g$): The effective gravity is $g_{\text{eff}} = g - g = 0$. In state of weightlessness, there is no hydrostatic pressure difference across the depth of the fluid, so no force pushes the water out. Thus, no water leaks out of the jar. S matches with 4.

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Conclusion

The correct matching is: P-1, Q-1, R-1, S-4

Correct Option: C