Question

Each minute, a special game warden's machine gun fires 220 , 12.6-g rubber bullets with a muzzle velocity of \(975 \mathrm{~m} / \mathrm{s}\). How many bullets must be fired at an \(84.7-\mathrm{kg}\) animal charging toward the warden at \(3.87 \mathrm{~m} / \mathrm{s}\) in order to stop the animal in its tracks? (Assume that the bullets travel horizontally and drop to the ground after striking the target.)

Short answer

To stop the animal in its tracks, approximately 27 bullets are required.

Step-by-step solution

Calculation of Animal's Momentum

Firstly, calculate the momentum of the charging animal. Momentum is the product of mass and velocity, so calculate it as \( Momentum = Mass_{animal} \times Velocity_{animal} = 84.7 kg \times 3.87 m/s = 327.869 kg \cdot m/s \)

Calculation of Bullet's Momentum

Next, calculate the momentum of a single bullet. Remember that the bullet's mass needs to be in kilograms, so convert the given mass from grams to kilograms. \(Mass_{bullet} = 12.6g = 12.6/1000 kg = 0.0126 kg \). Now calculate the bullet's momentum as \( Momentum_{bullet} = Mass_{bullet} \times Velocity_{bullet} = 0.0126 kg \times 975 m/s = 12.285 kg \cdot m/s \)

Number of Bullets Required

To stop the animal, the total momentum imparted by the bullets must equal the animal's momentum. Thus, calculate the number of bullets required as \(Number_{bullets} = Momentum_{animal} / Momentum_{bullet} = 327.869 kg \cdot m/s / 12.285 kg \cdot m/s \). Round this value up to the nearest whole number since we can't have fractional bullets.

Key concepts

Conservation of Momentum

In physics, the conservation of momentum is a fundamental principle that describes how the total momentum in a closed system remains constant over time, provided no external forces are acting on it. This law is essential in understanding various physical phenomena, especially in collisions or explosions. When applied to the problem of stopping a running animal using rubber bullets, we have to equate the animal's momentum with the momentum provided by the bullets.

To ensure these balanced forces:

• Calculate the momentum of the object we're stopping—here, the animal.
• Determine how the bullets' cumulative momentum can counteract the animal's momentum.

In this situation, for the animal to stop in its tracks, the forward momentum of the animal must be equal to the total momentum imparted in the opposite direction by the bullets. This aligns with the iconic physical concept that unless acted upon by an external force, momentum in motion will remain constant.

Linear Momentum

Linear momentum, often simply referred to as momentum, is the product of an object's mass and its velocity. It is a vector quantity, which means it has both magnitude and direction. In the exercise, the momentum of both the animal and the bullets were calculated.
The steps are straightforward:

• For the animal: Multiply its mass (\(84.7 ext{ kg}\)) by its velocity (\(3.87 ext{ m/s}\)). This gives a momentum value that must be overcome.
• For each bullet: The mass is converted from grams to kilograms, and then multiplied by the muzzle velocity (\(975 ext{ m/s}\)).

The formula is simple yet powerful:\[Momentum = Mass \times Velocity\]Understanding how to apply this formula helps in tackling a wide variety of physics problems involving moving objects.

Physics Calculations

Physics calculations often involve applying formulas accurately to derive meaningful results. In this exercise, we are linking physical laws with mathematical calculations to determine how many bullets are necessary to stop an animal.
To solve such a problem:

• Identify the quantities needed (mass and velocity in our case) and ensure units are consistent, converting when necessary (e.g., grams to kilograms).
• Precisely perform multiplication of these quantities to find momentum values.
• Use the equation \(Number_{bullets} = \frac{Momentum_{animal}}{Momentum_{bullet}}\) to determine how many bullets are needed.

The integrity of your calculations lies in precision and keen attention to detail, as demonstrated in rounding up to the nearest whole number of bullets since partial bullets don't exist in practice. Mastery in using these fundamental techniques is crucial for solving not only textbook problems but also real-world physics challenges.