A 14-cm diameter champagne bottle rests on its side on top of a frictionless horizontal table.Suddenly,the cork pops and the bottle slides backward,covering a distance of 22.0 cm, in 0.44 s. If the mass of the bottle is 500 times the mass of the cork,find the distance from the original position that the cork will land on the table.Neglect air resistance and assume that the cork is very small compared to the bottle.
A)3,000 cm
B)60 m
C)85 m
D)8.5 cm

Question

A 14-cm diameter champagne bottle rests on its side on top of a frictionless horizontal table.Suddenly,the cork pops and the bottle slides backward,covering a distance of 22.0 cm, in 0.44 s.  If the mass of the bottle is 500 times the mass of the cork,find the distance from the original position that the cork will land on the table.Neglect air resistance and assume that the cork is very small compared to the bottle.
A)3,000 cm
B)60 m
C)85 m
D)8.5 cm

Quizplus · Accepted Answer

The cork's distance can be found using the principle of conservation of momentum. Since the system is isolated and there's no external force acting on it, the momentum before the cork pops is equal to the momentum after it pops. Before the cork pops, the system (bottle + cork) is at rest, so the initial momentum is 0. After the cork pops, the momentum of the bottle and the cork must still sum to 0.Let $m$ be the mass of the cork, $v_c$ the velocity of the cork, $500m$ the mass of the bottle, and $v_b$ the velocity of the bottle. The conservation of momentum gives us:$$0 = m \cdot v_c + 500m \cdot v_b$$$$v_c = -500 \cdot v_b$$The negative sign indicates that the cork and the bottle move in opposite directions. We know the bottle moves 22.0 cm in 0.44 s, so its speed $v_b$ is:$$v_b = \frac{22.0\, \text{cm}}{0.44\, \text{s}} = 50\, \text{cm/s}$$Thus, the cork's speed $v_c$ is:$$v_c = -500 \cdot 50\, \text{cm/s} = -25,000\, \text{cm/s}$$Since air resistance is neglected, the horizontal velocity of the cork remains constant. The distance the cork travels, $d$, is then:$$d = v_c \cdot t$$To find $t$, we use the fact that the vertical motion of the cork (ignoring since it lands on the table) and the horizontal motion are independent. The cork and the bottle are initially at the same height, and the cork lands back on the table, so its vertical displacement is 0. However, since we're told to neglect air resistance and the problem simplifies to finding the horizontal distance without considering vertical motion explicitly, we focus on the horizontal component.Given the information, we don't have a direct way to calculate the time of flight for the cork based on vertical motion parameters (since vertical motion details like height or gravity's effect aren't provided or necessary for the calculation as framed). The problem setup implies focusing on the horizontal outcome, leveraging the conservation of momentum for the system's behavior.Given the setup and the information provided, the calculation directly leading to the cork's distance without integrating time explicitly from vertical motion parameters suggests an oversight in the direct application of the provided data. The correct approach involves recognizing the conservation of momentum to deduce the relative speeds and then applying this to understand the system's dynamics, but the final step to find the exact landing distance of the cork would typically require integrating both horizontal velocity and time, which in this context, seems to be indirectly addressed through the momentum conservation and the given data about the bottle's movement.The initial setup implies a direct calculation based on the provided speeds and the relationship between the bottle and cork's mass. However, without explicit vertical motion parameters or a clear path to integrate time based on such, the solution focuses on the horizontal dynamics governed by momentum conservation. The oversight in not detailing the vertical motion calculation step or the explicit integration of time based on vertical motion parameters reflects a simplification or an assumption in the problem's framing, focusing on the conservation of momentum and the direct relationship between the velocities of the cork and bottle.Given the misunderstanding in the explanation process, the correct approach to find the distance the cork travels, considering the conservation of momentum and the relationship between the velocities, would indeed involve understanding the system's dynamics based on the principles provided. However, the calculation of the exact distance without integrating time from vertical motion parameters directly seems to have been an oversight in the explanation process.

A 14-Cm Diameter Champagne Bottle Rests on Its Side on Top