Question

A railroad flatcar of weight \(W\) can roll without friction along a straight horizontal track. Initially, a man of weight \(w\) is standing on the car, which is moving to the right with speed \(v_{0}\). What is the change in velocity of the car if the man runs to the left (Fig. 6-22) so that his speed relative to the car is \(v_{\text {rel }}\) just before he jumps off at the left end?

Short answer

The change in velocity of the car would be \( Δv = \frac{mv_{\text{rel}}}{M} \).

Step-by-step solution

Initial momentum of the system

First, calculate the total initial momentum of the system. It consists of the man and the flatcar moving to the right with speed \(v_0\). The initial momentum, \(P_{\text{initial}}\), would therefore be: \(P_{\text{initial}}= (m+M)v_0\), where m is the mass of the man (whose weight is w), M is the mass of the flatcar (whose weight is W), and \(v_0\) is the initial velocity of the system.

Final momentum of the system

After the man jumps off, the final momentum of the system includes the man moving to the left with speed \(v_{\text{rel}}\) relative to the flatcar and the flatcar itself. However, we need to find out the speed of the man and the flatcar with respect to the ground. Let's write the final momentum, \(P_{\text{final}}\), as: \(P_{\text{final}} = mv_{\text{man}} + Mv_{\text{car}}\), where \(v_{\text{man}}\) is the velocity of the man with respect to the ground and \(v_{\text{car}}\) is the speed of the flatcar with respect to the ground.

Apply conservation of momentum

Since there are no external forces acting on the system, the total momentum of the system is conserved. From the conservation of momentum, we equate the initial and final momenta: \(P_{\text{initial}} = P_{\text{final}} \). Solving the equation, we get \( (m+M)v_0 = mv_{\text{man}} + Mv_{\text{car}} \). Now, it's clear that the man's speed relative to the ground, \(v_{\text{man}} = v_{\text{car}} - v_{\text{rel}} \). Plugging this result back into the equation gives us the change in velocity of the flatcar.

Key concepts

Momentum

When we hear the term 'momentum' in the context of physics, we're talking about the quantity of motion an object has. This quantity is dependent not just on the object's speed, but also on its mass. The formula to calculate momentum (\( p \) is the conventional symbol for momentum) is given by the equation: \( p = mv \) where \( m \) is the mass of the object and \( v \) is its velocity.

For example, if a baseball and a freight train are both moving at the same speed, the train has a considerably more momentum due to its much greater mass. This is why it takes much more effort to stop a train than a baseball!

It's crucial to understand that momentum is a vector, which means it has both a magnitude (how much) and a direction. When solving problems involving momentum, particularly when objects are moving in different directions, we need to be careful to account for the direction of each momentum vector. And when two objects interact, such as a man jumping off a moving flatcar, the total momentum before and after the interaction stays the same if no external forces act on the system. This principle is known as the conservation of momentum.

Classical Mechanics

Classical mechanics is a branch of physics that deals with the motion of objects that are much larger than atoms and move at much slower speeds than the speed of light. It's the framework within which most everyday phenomena can be understood. Within this framework, we make use of Newton's laws of motion, which describe the relationships between the forces applied to an object, its mass, and its acceleration.

Essential to classical mechanics are three core concepts: inertia, force, and conservation laws, including conservation of momentum, energy, and angular momentum. Inertia is the property of an object to resist changes in its state of motion, force is what causes an object to accelerate, and conservation laws describe quantities that remain constant throughout the physical processes.

The problem involving the man and the flatcar is an excellent example of classical mechanics at work. By investigating the system before and after the man jumps off, we can apply these principles to solve for the unknown variables such as the change in velocity of the flatcar.

Physics Problem Solving

Solving physics problems can often seem daunting, but breaking the problem into manageable steps is a key strategy. This process typically involves identifying the known and unknown variables, applying physical laws, and solving resulting equations.

When faced with a problem, start by carefully reading to understand what is being asked and what information is provided. The next step is to choose an appropriate frame of reference and identify the principles that apply to the problem – for our man and flatcar dilemma, this is conservation of momentum.

Using a systematic approach, as seen in the textbook solution, helps to clarify complex situations. Sketching a diagram can also be very beneficial. Once the principles are applied and equations are set up, it's a matter of solving for the unknowns. Last but not least, always check if your solution makes physical sense. Does the direction of the final velocities correspond with the initial conditions? Have you accounted for all the forces and masses involved?

Exercise Improvement Advice

For our current problem, we might also consider including air resistance or the friction between the man's feet and the flatcar as factors that could affect the final outcome.