Question

A bucket is mounted on a skateboard, which rolls across a horizontal road with no friction. Rain is falling vertically into the bucket. The bucket is filled with water, and the total mass of the skateboard, bucket, and water is \(M=10.0 \mathrm{~kg} .\) The rain enters the top of the bucket and simultaneously leaks out of a hole at the bottom of the bucket at equal rates of \(\lambda=0.100 \mathrm{~kg} / \mathrm{s}\). Initially, bucket and skateboard are moving at a speed of \(v_{0}\). How long will it take before the speed is reduced by half?

Short answer

Answer: It will not take any time for the speed to reduce by half because it never does. The motion will continue indefinitely with the speed remaining constant at \(v_0\), as there are no external forces acting on the system and the rainwater entering and leaving the bucket does so at an equal rate.

Step-by-step solution

Initial Momentum Calculation

Determine the momentum of the skateboard, bucket, and water system initially. Since the initial velocity is \(v_0\), the initial momentum is: \(p_0 = Mv_0\)

Momentum During Rainfall

During rain falling into the bucket and simultaneously leaking out of the hole, the mass remains constant. But the speed of the skateboard gradually decreases due to the forces acting upon it. Let's say the speed of the skateboard is \(v\) at a certain time \(t\). At this time, the momentum of the system is: \(p_t = Mv\)

Momentum Conservation

Since there is no external force acting on the system, the linear momentum of the system remains constant. So, \(p_0 = p_t\). Also, we want to find out when the speed is reduced by half, so \(v = \frac{v_0}{2}\). Equating the momentums: \(Mv_0 = M(\frac{v_0}{2})\)

Solving for Time

We can solve the momentum equation for time: \(v_0 = \frac{v_0}{2}\) This equation implies that the speed of the skateboard remains constant throughout the process. It never reduces by half as there are no external forces acting on it and as the rainwater entering and leaving the bucket does so at an equal rate. So, it will not take any time for the speed to reduce by half because it never does, and the motion will continue indefinitely with the speed remaining constant at \(v_0\).

Key concepts

Understanding Linear Momentum

Linear momentum is a fundamental concept in physics that measures the quantity of motion of a moving body. It is the product of an object's mass (\( m \)) and its velocity (\( v \)), represented as \( p = mv \). In simpler terms, it tells us how hard it would be to stop a moving object.

When dealing with problems in physics, such as a skateboard moving across a frictionless surface with a bucket on top collecting rain, momentum conservation plays a crucial role. By understanding that the total linear momentum of a closed system remains constant when there are no external forces, students can predict how the system will behave over time.

For example, in the absence of external forces, even as rain falls into the bucket, if the rain is leaking at the same rate, the system's mass remains constant. Therefore, the velocity of the skateboard with bucket must also remain constant to conserve momentum. This directly affects the solution to the problem as it implies the skateboard's speed will not change.

Exploring Forces and Motion in Physics

Forces and motion are core elements in physics, offering insight into how objects interact and move in our world. The laws of motion formulated by Sir Isaac Newton provide a framework to understand these concepts.

Motion pertains to the change in position of an object over time and is directly influenced by forces. Forces, which can be described as pushes or pulls, cause objects to accelerate, slow down, change direction, or remain in place. In the given exercise, the only forces at play are gravity (acting on the rain) and the leaking water's force.

Newton's Second Law

According to Newton's Second Law, when a force is applied to an object, it accelerates proportionally to the force and inversely to its mass (\( F = ma \)). However, since we're assuming a no-friction environment, the skateboard's horizontal motion is not affected by any external force, signifying no acceleration and, hence, a constant velocity.

The Rainfall and Leakage Problem

The rainfall and leakage problem in physics is an application of the conservation of momentum in a real-life context. It demonstrates how physics principles help us analyze and solve interesting scenarios beyond textbook examples.

In our problem, rain falls vertically into a moving bucket, and water exits the bucket at the same rate. This type of problem involves careful consideration of the system's mass and how external factors, like rain, influence it. Despite rain adding mass to the system, if there's simultaneous leakage at the same rate, the system's total mass doesn't change.

Practical Implications

This kind of analysis is highly useful in engineering and environmental studies, where the movement and balance of materials must be maintained. For students, the takeaway is that even when external elements are introduced into a system, such as rain into a bucket, if there is no net change in mass or external force, the system's momentum will remain constant, leading to motion that is predictable and constant over time.