Physicians perform needle biopsies to sample tissue from internal-organs. A spring-loaded gun shoots a hollow needle into the tissue; extracting the needle brings out the tissue core. A particular device uses 8.3 -mg needles that take 90 ms to stop in the tissue, which exerts a stopping force of \(41 \mathrm{mN}\). (a) Find the impulse imparted by the tissue. (b) How far into the tissue does the needle penetrate?

Impulse in physics is a fundamental concept directly connected to momentum. It is defined as the change in momentum of an object when it is acted upon by a force over a specific time interval. The impulse can be depicted by the equation:
\( Impulse = Force \times Time \).
In our needle biopsy example, as the spring-loaded gun shoots the needle, the tissue exerts a stopping force over the time it takes to stop the needle. The calculated impulse of \(3.69 \times 10^{-3} \, Ns\) allows us to understand the 'push' the tissue had to deliver to bring the needle to rest.
For students wanting to improve their understanding of impulse:

• Think of impulse as a 'push' or 'pull' acting over a period.
• Note that impulse is a vector quantity, carrying both magnitude and direction.
• Impulse is represented by the area under a force-time graph.

Grasping this will clarify why physicians can use it to determine how much effort (force over time) was required to stop the needle during a biopsy.

Momentum is a measure of the quantity of motion an object has and is calculated by multiplying an object's mass by its velocity. The principle of momentum change is integral to understanding how objects interact with forces.
\( Momentum = Mass \times Velocity \).
When the needle in the biopsy gun comes to a stop, the change in momentum is significant; it goes from a specific initial velocity to zero. This is directly proportional to the impulse, according to the equation:
\( Impulse = Change \, in \, Momentum \).
As such, using the impulse, we can backtrack to find the needle's initial velocity. For students, remember that:

• A change in momentum can result from a change in either mass, velocity, or both.
• In the absence of external forces, momentum is conserved.

These points are key to understanding the mechanics of how the needle behaves once it's been fired by the biopsy gun.

The equations of motion are a set of formulas that predict the future position and velocity of an object moving under constant acceleration. In the context of our needle biopsy example, they help predict how far the needle will penetrate the tissue.
One of these equations is represented as:
\( distance = 0.5 \times (Initial \, Velocity + Final \, Velocity) \times Time \).
Here, since the final velocity is zero, the equation simplifies to account for the initial velocity over the time interval of the needle's deceleration in the tissue. The equation helps us estimate the penetration depth to be 20 meters, but this unusually high value suggests a potential error. It is vital to ensure units are consistent and the equation applies to the scenario realistically. For students:

• Always check units to avoid errors.
• Understand that constant deceleration is akin to negative acceleration.
• Remember, equations of motion are valid only under constant acceleration or deceleration.

Understanding these concepts is essential not just for solving physics problems but also for appreciating the physical phenomena in real-world applications like medical procedures.