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Qmechanic ♦
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Consider a bucket of weight $W$, held in air by a jet stream of water from the ground (moving vertically upwards) at height $h$. The water is fired from the ground at the rate of $\mu$ (units kg/s) with an initial velocity of $v_0$. Find the maximum possible value of $h$. Find conditions to achieve the same.Consider a bucket of weight $W$, held in air by a jet stream of water from the ground (moving vertically upwards) at height $h$. The water is fired from the ground at the rate of $\mu$ (units kg/s) with an initial velocity of $v_0$. Find the maximum possible value of $h$. Find conditions to achieve the same.

(To observers, this is a problem from Kleppner,Kollenkow; An Introduction to Mechanics & Kollenkow: An Introduction to Mechanics).

What I did: Consider a differential mass element $dm$ near the bucket before hitting it. Its velocity is $v_f = \sqrt{v_0^2 - 2gh}$, so its momentum is $v_f dm$. Suppose after hitting the bucket it had a velocity of $v'$, and the time of contact was $\Delta t$. Since we must exert a force of $W$ upwards on the bucket, the rate-of-change of momentum of $dm$ is $W$. Clearly $dm(v' + v_f) = W\Delta t$, which in the limit $\Delta t \to 0$, becomes $W = \mu (v' + v_f)$. Thus $2gh = v_0^2 - \left(\frac{W}{\mu} - v'\right)^2 $. This raises a question. Setting $v' = \frac{W}{\mu}$, we can attain $h = v_0^2 / 2g$, but at that height, the water would have no velocity, so no momentum change.

What are my errors, and how to attack such problems (with continuous masses)? ( Please excuse me for any silly mistakes. I'm only a beginner :( ). A detailed solution which I can study would be extremely helpful.

Consider a bucket of weight $W$, held in air by a jet stream of water from the ground (moving vertically upwards) at height $h$. The water is fired from the ground at the rate of $\mu$ (units kg/s) with an initial velocity of $v_0$. Find the maximum possible value of $h$. Find conditions to achieve the same.

(To observers, this is a problem from Kleppner,Kollenkow; An Introduction to Mechanics).

What I did: Consider a differential mass element $dm$ near the bucket before hitting it. Its velocity is $v_f = \sqrt{v_0^2 - 2gh}$, so its momentum is $v_f dm$. Suppose after hitting the bucket it had a velocity of $v'$, and the time of contact was $\Delta t$. Since we must exert a force of $W$ upwards on the bucket, the rate-of-change of momentum of $dm$ is $W$. Clearly $dm(v' + v_f) = W\Delta t$, which in the limit $\Delta t \to 0$, becomes $W = \mu (v' + v_f)$. Thus $2gh = v_0^2 - \left(\frac{W}{\mu} - v'\right)^2 $. This raises a question. Setting $v' = \frac{W}{\mu}$, we can attain $h = v_0^2 / 2g$, but at that height, the water would have no velocity, so no momentum change.

What are my errors, and how to attack such problems (with continuous masses)? ( Please excuse me for any silly mistakes. I'm only a beginner :( ). A detailed solution which I can study would be extremely helpful.

Consider a bucket of weight $W$, held in air by a jet stream of water from the ground (moving vertically upwards) at height $h$. The water is fired from the ground at the rate of $\mu$ (units kg/s) with an initial velocity of $v_0$. Find the maximum possible value of $h$. Find conditions to achieve the same.

(To observers, this is a problem from Kleppner & Kollenkow: An Introduction to Mechanics).

What I did: Consider a differential mass element $dm$ near the bucket before hitting it. Its velocity is $v_f = \sqrt{v_0^2 - 2gh}$, so its momentum is $v_f dm$. Suppose after hitting the bucket it had a velocity of $v'$, and the time of contact was $\Delta t$. Since we must exert a force of $W$ upwards on the bucket, the rate-of-change of momentum of $dm$ is $W$. Clearly $dm(v' + v_f) = W\Delta t$, which in the limit $\Delta t \to 0$, becomes $W = \mu (v' + v_f)$. Thus $2gh = v_0^2 - \left(\frac{W}{\mu} - v'\right)^2 $. This raises a question. Setting $v' = \frac{W}{\mu}$, we can attain $h = v_0^2 / 2g$, but at that height, the water would have no velocity, so no momentum change.

What are my errors, and how to attack such problems (with continuous masses)? ( Please excuse me for any silly mistakes. I'm only a beginner :( ). A detailed solution which I can study would be extremely helpful.

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RandomStudent
RandomStudent
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Weight held by jet stream of water

Consider a bucket of weight $W$, held in air by a jet stream of water from the ground (moving vertically upwards) at height $h$. The water is fired from the ground at the rate of $\mu$ (units kg/s) with an initial velocity of $v_0$. Find the maximum possible value of $h$. Find conditions to achieve the same.

(To observers, this is a problem from Kleppner,Kollenkow; An Introduction to Mechanics).

What I did: Consider a differential mass element $dm$ near the bucket before hitting it. Its velocity is $v_f = \sqrt{v_0^2 - 2gh}$, so its momentum is $v_f dm$. Suppose after hitting the bucket it had a velocity of $v'$, and the time of contact was $\Delta t$. Since we must exert a force of $W$ upwards on the bucket, the rate-of-change of momentum of $dm$ is $W$. Clearly $dm(v' + v_f) = W\Delta t$, which in the limit $\Delta t \to 0$, becomes $W = \mu (v' + v_f)$. Thus $2gh = v_0^2 - \left(\frac{W}{\mu} - v'\right)^2 $. This raises a question. Setting $v' = \frac{W}{\mu}$, we can attain $h = v_0^2 / 2g$, but at that height, the water would have no velocity, so no momentum change.

What are my errors, and how to attack such problems (with continuous masses)? ( Please excuse me for any silly mistakes. I'm only a beginner :( ). A detailed solution which I can study would be extremely helpful.