Question

A railway flat car is rushing along a level frictionless track at a speed of \(45 \mathrm{~m} / \mathrm{s}\). Mounted on the car and aimed forward is a cannon that fires \(65-\mathrm{kg}\) cannon balls with a muzzle speed of \(625 \mathrm{~m} / \mathrm{s}\). The total mass of the car, the cannon, and the large supply of cannon balls on the car is \(3500 \mathrm{~kg}\). How many cannon balls must be fired to bring the car as close to rest as possible?

Short answer

Four cannonballs must be fired to bring the car as close to rest as possible.

Step-by-step solution

Understand the initial momentum

The initial momentum of this system can be calculated by multiplying the total mass of the system by their initial velocity. Hence, initial momentum = \( mass_{total} × velocity_{initial} = 3500 kg × 45 m/s = 157500 kg·m/s \).

Understand the final momentum

If we define forward as the positive direction, we can see that the final momentum of the system, following firing a cannonball, is composed of the momentum of the car-cannon system and the cannonball. Hence, it can be defined as: Final momentum = \( mass_{car-cannon} × velocity_{car-cannon} ) + ( mass_{cannonball} × velocity_{cannonball} ) . \) Assuming that the car comes as close to rest as possible, we can set its final forward velocity to 0, hence, Final momentum = \( 0 + ( 65 kg × -625 m/s ) = -40625 kg·m/s \), where we've taken the velocity of the ball as negative since it is going in the negative or backwards direction with respect to the car.

Evoke the Conservation of Momentum Law

According to the conservation of momentum, the total initial momentum should equal to the total final momentum. Setting these equal we get: \( 157500 kg·m/s = n × -40625 kg·m/s\). Solve for n to get number of cannonballs.

Compute the number of cannonballs

Solving for n we get: \( n = 157500 kg·m/s / -40625 kg·m/s = -3.88 \), n should be a whole number, since we cannot fire a fraction of cannonball. From the context, the negative sign means cannonballs are fired in the opposite direction of the initial motion of the car, thus we take the absolute value and then round up. Hence, the number of cannonballs to be fired is 4.

Key concepts

Physics Problem Solving

In solving physics problems, especially those involving motion and forces, it's crucial to have a solid understanding of the foundational principles. With this railway flat car problem, we're utilising the conservation of momentum as a tool to find out how the car can be brought to a halt.
First, properly identify what you need to solve, and what principles apply. Here, we're focusing on momentum, an essential concept in the physics of motion.
You start by collecting all the given data: the masses and velocities involved. Next, you calculate the initial and final momentum of the system, remembering the directions of forces and movements matter. Clear thought processes and setting equations step by step ensure you correctly apply physics laws such as conservation of momentum.

• Understand what is given and what needs to be found.
• Apply relevant principles like momentum conservation.
• Use systematic steps to calculate and solve.

Momentum in Physics

Momentum is a fundamental concept in physics. It describes the motion of an object and is calculated as the product of an object's mass and velocity, represented as \( p = m \cdot v \).
In this exercise, the flat car and cannonballs system begins with momentum from the car moving forward.
When a cannonball is fired, it gains backward momentum. This momentum affects the remaining system (the car and cannon), causing it to adjust its motion. The conservation of momentum principle tells us that the total momentum before and after an event must remain constant, provided no external forces act.

• Momentum is mass times velocity.
• Initial momentum = Total mass \( \times \) Initial velocity.
• Final momentum accounts for all parts of the system post-event.

Physics of Motion

The physics of motion deals with how objects move through space and time, influenced by forces. In this railway flat car problem, all three mutually interact to demonstrate motion.
Remember that forces and motion are deeply interconnected—Newton's laws come into play. Here, the conservation of momentum simplifies our calculations, especially on frictionless surfaces where we don't worry about additional forces slowing down the car.

• Object motion is evaluated using Newton's Laws.
• Conservation of momentum governs the system's behavior when firing cannonballs.
• Frictionless surfaces simplify our equations, only relying on initial and final states.

By understanding these core concepts within physics, you can dissect and solve even complex system-based problems such as this with clarity.