Problem 28 - University Physics with Modern Physics

Question

A .22-caliber rifle bullet traveling at 350 m/s strikes a large tree and penetrates it to a depth of 0.130 m. The mass of the bullet is 1.80 g. Assume a constant retarding force. (a) How much time is required for the bullet to stop? (b) What force, in newtons, does the tree exert on the bullet?

Step-by-Step Solution

Answer

(a) 7.43 × 10⁻⁴ s (b) 1.35 × 10³ N

Step 1: Convert Mass to Kilograms

The mass of the bullet is given as 1.80 grams. To use standard SI units, convert this mass to kilograms.Given: \[m = 1.80\, \text{g} = 1.80 \times 10^{-3} \text{ kg}\]

Step 2: Use Kinematic Equation to Find Acceleration

To find the time required for the bullet to stop, we first need to determine the acceleration. Use the kinematic equation that relates initial velocity, final velocity, acceleration, and distance: \[v_f^2 = v_i^2 + 2a d\]Where:- Initial velocity \(v_i = 350 \text{ m/s}\)- Final velocity \(v_f = 0 \text{ m/s}\)- Distance \(d = 0.130 \text{ m}\)Re-arrange to solve for acceleration \(a\):\[0 = (350)^2 + 2\times a\times 0.130\]Solve for \(a\):\[a = -\frac{(350)^2}{2 \times 0.130}\]Calculate \(a\).

Step 3: Calculate Time Using Acceleration

Now that we have acceleration, use it to find the time \(t\) using the following kinematic formula: \[v_f = v_i + a t\]Rearrange to solve for \(t\):\[t = \frac{v_f - v_i}{a} = \frac{0 - 350}{a}\]Substitute the calculated value of \(a\) into this equation to find \(t\).

Step 4: Calculate Force Using Acceleration and Mass

Once acceleration is determined, calculate the force exerted by the tree using Newton's Second Law:\[F = m a\]Where:- \(F\) is the force exerted by the tree- \(m = 1.80 \times 10^{-3} \text{ kg}\) is the mass of the bullet- \(a\) is the acceleration calculated in the previous stepsSubstitute the values to find \(F\) in Newtons.

Key Concepts

Constant Acceleration

When we talk about constant acceleration, we refer to a steady change in velocity over time. In the case of the rifle bullet penetrating the tree, this means that as soon as the bullet makes contact with the tree, it slows down at a uniform rate until it stops completely.
This constant deceleration (which is just acceleration working in the opposite direction of motion) simplifies calculations, as we can use specific kinematic formulas designed for such conditions.

• In our scenario: the bullet's velocity reduces to zero as it penetrates 0.130 meters into the tree.
• Because the force slowing down the bullet is consistent, its rate of slowdown is also constant.

This makes it easier to predict the bullet's behavior using formulas for constant acceleration.

Newton's Second Law

Newton's Second Law is central to understanding motion and forces. It tells us that acceleration is produced when a force acts on a mass, expressed in the equation: \[ F = m \times a \]Here, \(F\) is the force, \(m\) is the mass, and \(a\) is the acceleration. This law helps us calculate the force the tree exerts on the bullet using the mass of the bullet and its acceleration.
To find this force, we first determine the bullet's acceleration due to the retarding force of the tree. Once we know how quickly the bullet is slowing down, we can use its mass to find out the exact force being applied by the tree to stop it.

• Knowing \(a\), the deceleration, and \(m\), the mass of 1.80 grams (converted to kilograms), the force comes out in Newtons.
• The force is crucial for understanding how the bullet interacts with the tree and vice versa.

Kinematic Equations

Kinematic equations are incredibly useful tools in physics. They help analyze motion, especially when acceleration is constant. There are several equations, but one commonly used is:\[ v_f^2 = v_i^2 + 2a d \]In this specific exercise, the kinematic equation allows us to calculate the bullet’s deceleration without directly knowing the time factor initially. Knowing the distance of 0.130 meters, the initial velocity of 350 m/s, and the final velocity being zero, we rearrange the formula to solve for \(a\), acceleration:
- Substitute known values to find \(a\), and then use another kinematic formula:\[ v_f = v_i + a t \] To solve for \(t\), the time it takes for the bullet to stop.

• These equations only apply because the bullet’s acceleration is constant due to the uniform force applied by the tree.
• The calculated time gives insight into how quickly the bullet is halted during its penetration.

Using kinematic equations helps bridge the gap between knowing how far the bullet travels into the tree and understanding the time and force aspects of the scenario.