Animated Solution for Physics - Work, Energy, and Power: A bullet fired into a fixed target loses half of its velocity after penetrating 3 cm. How much further it will penetrate before coming to rest, assuming that it faces constant resistance to motion?

Imagine a high-speed bullet slamming into a thick, fixed wooden target. As it penetrates, the wood exerts a constant resistive force, acting like a powerful brake and slowing the bullet down. This classic physics problem asks us to determine how much further the bullet will travel after it has already lost half its velocity in the first $3\text{ cm}$.

Work-Energy Theorem

To connect the resistive force, the penetration distance, and the changing velocity, the Work-Energy Theorem is our best friend. It states that the net work done on an object equals its change in kinetic energy:

Since the resistive force $F$ acts opposite to the bullet's motion, the work done is negative: $W=-F\cdot x$.

The First 3 cm

Let the initial velocity of the bullet be $v_0$. After penetrating $x=3\text{ cm}$, its velocity drops to $\frac{v_0}{2}$. Let's apply the theorem to this first phase:

Squaring the half-velocity gives us a factor of $\frac{1}{4}$. Subtracting the initial kinetic energy leaves us with:

The minus signs and the $3$s cancel out beautifully, giving us an expression for the constant resistive force:

The Final Stretch

Now, the bullet continues to push through the wood for some unknown further distance, let's call it $s$, until it finally comes to a complete stop ($v=0$). We apply the Work-Energy Theorem one more time, just for this final stretch. The initial velocity for this phase is $\frac{v_0}{2}$, and the final velocity is $0$.

Now for the grand finale. We substitute the expression for $F$ we found earlier into this new equation:

Look at that! The minus signs cancel, and the entire $\frac{1}{8}m{v}_0^2$ term cancels out perfectly on both sides. We are left with:

So, the bullet will travel exactly $1\text{ cm}$ further before the wood completely stops it. It's a beautiful and elegant application of energy conservation!