Skaters' Collision: Calculating Final Velocity & Momentum

Hey guys! Let's dive into a classic physics problem, perfect for understanding momentum and velocity, inspired by a Fuvest-SP exam question. We'll break down the scenario step-by-step, making sure it's crystal clear. So, here we go!

Understanding the Problem

Alright, imagine this: Two skaters, of equal mass, are gliding along a straight path. The first skater is cruising at 1.5 m/s, while the second one, the faster skater, is zipping along at 3.5 m/s, chasing after the first. Here’s the kicker: the faster skater catches up, leaps vertically, and grabs onto the back of the slower skater. What happens next? How do we figure out their combined velocity after this awesome mid-air grab?

This problem is all about the principle of conservation of momentum. In a closed system (which, for our purposes, we can assume this is), the total momentum before an event equals the total momentum after the event. Momentum (p) is simply the mass (m) of an object multiplied by its velocity (v), or p = mv. So, before the collision (the grab, in this case), we have the momentum of both skaters. After the collision, we have the combined mass of the skaters moving together at a new velocity. Our goal is to find this final velocity.

To solve this, we'll use the formula for conservation of momentum: m₁v₁ + m₂v₂ = (m₁ + m₂)v_final. Where:

• m₁ is the mass of the first skater.
• v₁ is the velocity of the first skater.
• m₂ is the mass of the second skater.
• v₂ is the velocity of the second skater.
• v_final is the final velocity of the combined skaters.

Since the skaters have the same mass, we can simply call their mass 'm'. Let's get started with the calculation, shall we?

We know that the mass of both skaters is the same, let's call it 'm'. The first skater has a velocity (v₁) of 1.5 m/s, and the second skater has a velocity (v₂) of 3.5 m/s. The equation becomes: m * 1.5 + m * 3.5 = (m + m) * v_final. Simplifying this will lead us to the final velocity.

Setting Up the Calculation: The Physics Breakdown

Okay, before we jump into the math, let's make sure we've got our physics straight. The key concept here is the conservation of momentum. This fundamental principle of physics states that in a closed system (where no external forces are acting), the total momentum remains constant. Think of it like this: the total 'oomph' of the system before the skaters collide is the same as the total 'oomph' after they're stuck together. No 'oomph' is created or destroyed; it's just redistributed.

Now, how do we quantify 'oomph'? That's where momentum comes in. Momentum is a measure of an object's mass in motion. The more mass an object has, and the faster it's moving, the more momentum it has. The formula for momentum (p) is quite simple: p = mv, where 'm' is mass and 'v' is velocity. Velocity, remember, is speed with a direction.

In our skater scenario, we have two objects (the skaters) interacting. Before the collision (the grab), each skater has their own momentum. After the collision, they become a single combined mass moving together with a new velocity. The total momentum before the grab is the sum of the individual momenta of each skater. The total momentum after the grab is the combined mass of both skaters multiplied by their new, shared velocity.

Since no external forces are significantly affecting the skaters (like friction, which we're usually told to ignore in these ideal problems), we can assume the system is closed. This means the total momentum before the grab must equal the total momentum after the grab. This allows us to set up an equation, which we'll use to solve for the final velocity (v_final).

Let's assume the mass of each skater is 'm'. The initial momentum of the first skater is m * 1.5 m/s. The initial momentum of the second skater is m * 3.5 m/s. After the collision, the combined mass is 2m (since they're stuck together), and their final velocity is what we're trying to find (v_final). So, the equation becomes:

(m * 1.5) + (m * 3.5) = (2m) * v_final

From here, the math is straightforward. The most important part is understanding the why behind the math – the physics principles at play!

Step-by-Step Solution: Crunching the Numbers

Alright, let’s get our hands dirty with some calculations! We've already laid the groundwork by understanding the problem and setting up our equation based on the conservation of momentum. Now, it's time to crunch the numbers and find the final velocity of the two skaters.

Here’s a recap of our equation:

(m * 1.5 m/s) + (m * 3.5 m/s) = (2m) * v_final

Notice that the mass (m) appears in every term of the equation. This means we can simplify things by dividing every term by 'm'. This effectively cancels out the mass, leaving us with a simpler equation that only involves the velocities:

1. 5 m/s + 3.5 m/s = 2 * v_final

Now, let's combine the velocities on the left side of the equation:

1. 5 + 3.5 = 5.0 m/s

So, we have:

5. 0 m/s = 2 * v_final

To find the final velocity (v_final), we need to isolate it. We do this by dividing both sides of the equation by 2:

v_final = 5.0 m/s / 2 v_final = 2.5 m/s

And there you have it! The final velocity of the two skaters after the faster skater grabs onto the back of the slower skater is 2.5 m/s. This makes intuitive sense. The faster skater's momentum contributes to increasing the combined velocity, but since the mass has doubled, the final velocity is a value that falls between the two initial velocities.

This simple example perfectly illustrates the conservation of momentum. It's a fundamental concept in physics that's applicable in a wide range of scenarios, from collisions of cars to the movement of planets!

Conclusion: Key Takeaways and Insights

So, what did we learn from this skater showdown? Well, besides the fact that physics problems can be kinda fun, we reinforced the following key concepts:

• Conservation of Momentum: This is the big one! In a closed system, the total momentum before a collision (or interaction) equals the total momentum after the collision. This principle is a cornerstone of physics and applies to everything from billiard balls to rocket launches.
• Momentum: It's a measure of mass in motion (p = mv). The more massive an object is, or the faster it's moving, the more momentum it has. Momentum is a vector quantity, meaning it has both magnitude and direction.
• Elastic vs. Inelastic Collisions: This problem deals with an inelastic collision. In an inelastic collision, kinetic energy is not conserved. Some kinetic energy is converted into other forms, such as heat or sound (the sound of the skaters colliding, maybe!). In contrast, in an elastic collision (like two billiard balls hitting each other), kinetic energy is conserved.
• Problem-Solving Strategy: We broke down the problem into manageable steps: understanding the scenario, identifying the relevant physics principles, setting up the equation, solving for the unknown, and checking our answer. This is a generally applicable strategy for tackling any physics problem.

I hope this explanation was helpful, guys! Remember, the best way to master physics is to practice. Try changing the initial velocities or masses of the skaters in the problem and see how it affects the final velocity. That will allow you to deepen your understanding even more. Keep practicing, and you'll become a physics pro in no time! Until next time, keep those physics questions coming, and keep on learning. Bye for now!