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Problem Sets:
Problem
There is a cup filled with water. A pingpong is floating on top of it. The pingpong is pushed to the bottom of the cup with pingpong fully submerged. At $t=0$, the pingpong is let go, and it goes up and jumps out of the water before coming back down.
- Plot the acceleration of the Pingpong as a function of the time qualitiatively.
- Plot the pressure of the water on the bottom of the cup as a function of time qualitatively.
Hints
Hint 1:
[Q1] Plot the acceleration of the Pingpong as a function of the time qualitatively.
1. Key Concepts and Derivation The motion of the pingpong ball is governed by Newton's second law, $F_{net} = ma$. The net force is the sum of gravity ($F_g = mg$, downwards), the buoyant force ($F_B = \rho_w g V_{submerged}$, upwards), and the drag force ($F_D$, opposite to velocity). We define the upward direction as positive.
The acceleration of the ball is:
$$a = \frac{F_{net}}{m} = \frac{F_B - F_g - F_D}{m}$$We analyze the motion in three distinct phases:
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Phase 1: Fully submerged ($0 \le t < t_1$) The ball is released from rest ($v=0, F_D=0$). The buoyant force is maximum and constant ($F_B = \rho_w g V_{ball}$). The initial acceleration is at its maximum positive value:
$$a_{max} = \frac{\rho_w g V_{ball} - mg}{m} = \left(\frac{\rho_w}{\rho_{ball}} - 1\right)g$$As the ball's upward velocity $v$ increases, the downward drag force $F_D$ increases, causing the net upward force and thus the acceleration $a$ to decrease.
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Phase 2: Emerging from water ($t_1 \le t < t_2$) As the ball breaks the surface, the submerged volume $V_{submerged}$ decreases rapidly, causing the buoyant force $F_B$ to plummet. The net force quickly becomes negative (dominated by gravity and drag), resulting in a large downward (negative) acceleration.
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Phase 3: In the air ($t \ge t_2$) Once the ball completely leaves the water, the buoyant and water drag forces vanish instantly. The only significant force is gravity (ignoring air drag for simplicity). The acceleration becomes constant and negative.
$$a = \frac{-F_g}{m} = -g$$The transition from Phase 2 to Phase 3 is a discontinuity. Just before leaving the water, the acceleration is a large negative value due to water drag. Just after leaving, it jumps up to $-g$.
2. Qualitative Plot The acceleration-time graph shows:
- Starts at a large positive maximum $a_{max}$ at $t=0$.
- Decreases as the ball moves up through the water.
- Drops sharply as the ball emerges, crossing zero and becoming strongly negative.
- At time $t_2$ when the ball leaves the water, the acceleration discontinuously jumps up to the constant value of $-g$.
- Remains at $-g$ for the remainder of its flight.