Question

You are trying to improve your shooting skills by shooting at a can on top of a fence post. You miss the can, and the bullet, moving at \(200 . \mathrm{m} / \mathrm{s},\) is embedded \(1.5 \mathrm{~cm}\) into the post when it comes to a stop. If constant acceleration is assumed, how long does it take for the bullet to stop?

Short answer

Answer: It takes approximately 0.00075 seconds for the bullet to stop after hitting the fence post.

Step-by-step solution

Identify the equations of motion

We can use the following equation of motion to solve for the time it takes the bullet to stop: v = u + a*t where: - v is the final velocity (0 m/s, since the bullet stops) - u is the initial velocity (200 m/s) - a is the acceleration - t is the time we want to find. But first, we need to find the constant acceleration (a). We can use the equation: v^2 = u^2 + 2*a*s where: - s is the distance the bullet travels into the post (0.015 m).

Calculate the constant acceleration

Using the second equation, v^2 = u^2 + 2*a*s, and given values, we get: 0^2 = 200^2 + 2*a*0.015 Solve for a: a = -(200^2) / (2*0.015) a = -266666.67 m/s²

Calculate the time it takes for the bullet to stop

Now, we'll use the first equation of motion, v = u + a*t, to find the time it takes the bullet to stop: 0 = 200 + (-266666.67)*t Solve for t: t = -200 / -266666.67 t = 0.00075 s So, it takes approximately 0.00075 seconds for the bullet to stop after hitting the fence post.

Key concepts

Equations of Motion

The equations of motion are fundamental formulas that describe the motion of objects under the influence of forces. These equations allow us to find different motion parameters, such as time, velocity, and displacement. In kinematics, especially under constant acceleration, two key equations frequently used are:

• \( v = u + a \times t \)
• \( v^2 = u^2 + 2 \times a \times s \)

In the exercise about shooting at a can, we apply these equations to deduce both the acceleration and the time it takes for the bullet to come to a rest. This demonstrates how powerful these equations can be when trying to solve motion-related problems.

Constant Acceleration

Constant acceleration implies that the rate of velocity change is uniform throughout the motion. This means the object speeds up or slows down at a steady rate. This is a crucial assumption in many physics problems, including the bullet embedding scenario.

When the bullet enters the post, it decelerates at a constant rate until it stops. To calculate this rate, or acceleration, we use the equation of motion:

• \( v^2 = u^2 + 2 \times a \times s \)

Here, by rearranging and substituting in known values, we can find the acceleration:\[ a = \frac{{-u^2}}{{2 \times s}} = \frac{{-200^2}}{{2 \times 0.015}} \ = -266666.67 \, \text{m/s}^2 \]This negative sign indicates the bullet is decelerating. Understanding constant acceleration is key to solving many real-world kinematics problems where forces cause objects to change velocity at a steady rate.

Initial Velocity

Initial velocity is the speed of an object before any forces are applied to it after a specific event starts. In our scenario, the initial velocity of the bullet just before impacting the post is crucial for calculations.

The bullet's given initial velocity is \(200 \, \text{m/s}\). This represents how fast it was moving before embedding itself into the post and is used in both equations of motion to determine acceleration and stopping time. Understanding initial velocity allows you to track changes in motion over time.

Final Velocity

Final velocity refers to the speed of an object at the end of the motion or after a specific interaction. In the bullet's scenario, its final velocity is \(0 \, \text{m/s}\), indicating it comes to a complete stop once embedded into the post.

• This is a vital parameter in motion equations, notably: \( v = u + a \times t \)
• Here, \(v\) is set to zero, reflecting a full stop which aids significantly in solving for time and acceleration.

Recognizing when and how final velocity is achieved is essential for evaluating an object's complete motion pathway, especially under constant acceleration conditions. This value directly reflects the effectiveness of the opposing forces, like friction or impact, in ceasing motion.