Problem 22 - College Physics

Question

A rifle with a weight of \(30 \mathrm{~N}\) fires a \(5.0\)-g bullet with a speed of \(300 \mathrm{~m} / \mathrm{s}\). (a) Find the recoil speed of the rifle. (b) If a \(700-\mathrm{N}\) man holds the rifle firmly against his shoulder, find the recoil speed of the man and rifle.

Step-by-Step Solution

Answer

The recoil speed of the rifle is approximately \(0.051 \) m/s, and the recoil speed of the man with the rifle is approximately \(0.0021 \) m/s.

Step 1: Define known variables and calculate momentum of the bullet

The weight of the rifle is 30 N, and the weight of the bullet is 5 g and has a speed of 300 m/s. We convert the weight of the bullet in kg, therefore \(0.005 \) kg. We calculate the momentum of the bullet using the formula \(Momentum = m_{bullet} * v_{bullet}\) which will give us \(1.5 \) kg*m/s.

Step 2: Calculate the recoil speed of the rifle

We calculate the recoil speed of the rifle applying the law of conservation of momentum. As there are no external forces, the momentum before and after firing the bullet must be equal. Therefore, \(m_{rifle} * v_{rifle} = m_{bullet} * v_{bullet}\). We convert the weight of the rifle into kilograms by dividing by 9.8 (the gravity acceleration). After calculating, we find that the recoil speed of the rifle is \( \frac{Momentum_{bullet}}{m_{rifle}} \approx 0.051 \) m/s.

Step 3: Calculate the recoil speed of the man and the rifle

Now we include the man holding the rifle in the momentum system. The total mass is now the mass of the man plus the mass of the rifle. We convert the weight of the man into mass. Using the conservation of momentum again, \(v_{man+rifle} = \frac{Momentum_{bullet}}{m_{man+rifle}}\). Therefore, the speed of the man and the rifle is about \(0.0021\) m/s.

Key Concepts

Conservation of Momentum

Imagine you're watching a game of pool. When the cue ball hits another, the second ball moves while the cue ball slows down. This happens because of a fundamental principle called the conservation of momentum. In physics, this principle states that if no external forces are acting on a system, the total momentum of that system remains constant.

An external force could be anything from a hand pushing an object to gravity acting on it. In our exercise, we're looking at a rifle firing a bullet. When a bullet is fired, the gun pushes the bullet forward, and in turn, the bullet pushes the gun backward. This backward 'push' is known as the recoil. Despite these internal forces, no external forces are acting on the bullet-gun system, so the total momentum before the shot (which is zero, as neither the gun nor the bullet are moving) must equal the total momentum after the shot.

Therefore, the momentum gained by the bullet will be equal in magnitude and opposite in direction to the momentum gained by the rifle, which explains why the rifle recoils backwards when the bullet is fired forward.

Momentum Calculation

Momentum might make you think of someone with a lot of drive or determination, but in physics, it's a bit more precise. The momentum of an object is calculated by the product of its mass and velocity. This can be remembered easily by the equation: \( Momentum = mass \times velocity \).

It's like mass and velocity are having a dance, and their dance move—their combined effect—is what we call momentum. In the context of the textbook exercise, when a bullet is fired from a rifle, the bullet gains momentum due to its speed. Since its mass is small, you might think its momentum is also small, but because its velocity is so high, even such a tiny bullet can pack a substantial momentum punch. To calculate the momentum of the bullet, we multiply its mass (after converting it from grams to kilograms) by its velocity. This gives us a clear number for how much 'oomph' the bullet leaves the rifle with, which is essential for figuring out just how much the rifle will recoil.

Newton's Second Law

Newton's second law is one of the cornerstones of physics, and it's often summed up as \( F=ma \), which means force equals mass times acceleration. Think of it as the universe's way of telling us how things move and why. When you apply a force to an object, it accelerates, and that acceleration depends on the object’s mass and the amount of force applied.

But how does this relate to our exercise? Well, when the bullet accelerates out of the rifle, there’s a force at play. The rifle experiences this force in the opposite direction, which causes it to accelerate backward, albeit much less noticeably due to its greater mass relative to the bullet. This is why both a rifle and a man would recoil at different speeds when firing a bullet of a given momentum: The mass of the rifle alone is less than the combined mass of the man and the rifle, so the rifle alone would have a higher recoil speed than the man and rifle together.