Skater Throws Stone: Physics Speed Calculation

Hey guys! Ever wondered about the physics behind everyday actions, like a skater throwing something? Today, we're diving deep into a classic physics problem that’ll make you think. We've got a skater, a stone, and some seriously cool physics principles at play. So, buckle up, grab your notebooks, and let's break down how to figure out the skater's speed after they launch that stone. This isn't just about numbers; it's about understanding motion, momentum, and how forces interact in the real world. We'll get into the nitty-gritty of calculating velocities and understanding the conservation of momentum, which is a super important concept in physics. Get ready to flex those brain muscles because this problem is going to be a fun ride!

Understanding the Physics at Play

Alright, let's talk physics, guys! The core concept we need to get our heads around for this problem is the conservation of momentum. In a closed system, the total momentum before an event (like the skater throwing the stone) is equal to the total momentum after the event. Momentum itself is pretty straightforward to calculate: it's just the mass of an object multiplied by its velocity (p = mv). So, before the throw, let's assume our skater and the stone are at rest, meaning their initial momentum is zero. When the skater throws the stone horizontally, they exert a force on it, and by Newton's third law (for every action, there is an equal and opposite reaction), the stone exerts an equal and opposite force back on the skater. This force causes both the stone and the skater to move. The stone moves forward, and the skater moves backward. The trick here is that the total momentum of the skater-stone system must remain zero after the throw. This means the momentum of the stone moving forward must be exactly cancelled out by the momentum of the skater moving backward.

We're given that the stone has a mass of 1 kg and it travels 20 meters horizontally in 4 seconds. This allows us to calculate the stone's horizontal velocity. Velocity is simply distance divided by time. So, the stone's velocity ($v_{stone}$) will be 20 meters / 4 seconds, which equals 5 meters per second. Now, here's where the conservation of momentum really shines. Since the initial momentum was zero, the final momentum must also be zero. The momentum of the stone is its mass (1 kg) multiplied by its velocity (5 m/s), giving us a momentum of 5 kgm/s. To keep the total momentum at zero, the skater must have an equal and opposite momentum. The skater's mass is 75 kg. Let's call the skater's final velocity $v_{skater}$. The momentum of the skater is $m_{skater} * v_{skater}$, which is 75 kg * $v_{skater}$. For the total momentum to be zero, the momentum of the stone and the momentum of the skater must be equal in magnitude but opposite in direction. So, $p_{stone} + p_{skater} = 0$. This means 5 kgm/s + (75 kg * $v_{skater}$) = 0. Solving for $v_{skater}$, we get $v_{skater}$ = -5 kg*m/s / 75 kg. The negative sign simply indicates that the skater moves in the opposite direction to the stone. So, the skater's speed is 5/75 m/s, which simplifies to approximately 0.067 m/s. Pretty neat, huh? It shows that even a small stone can result in a noticeable (though small!) movement for a much heavier skater.

Calculating the Stone's Velocity

Let's get down to brass tacks and calculate the stone's velocity, because this is the key to unlocking the skater's speed. We're given some juicy details: the stone's mass ($m_{stone}$) is 1 kg, it travels a horizontal distance ($\Delta x_{stone}$) of 20 meters, and this journey takes a time ($\Delta t$) of 4 seconds. To find the velocity, we use the fundamental formula: velocity = distance / time. It’s one of those basic, yet incredibly powerful, equations in physics that helps us describe how things move. So, for the stone, its horizontal velocity ($v_{stone}$) is calculated as:

$$v_{stone} = \frac{\Delta x_{stone}}{\Delta t}$$

Plugging in our numbers:

$$v_{stone} = \frac{20 \text{ meters}}{4 \text{ seconds}}$$

$$v_{stone} = 5 \text{ m/s}$$

Boom! Just like that, we know the stone is zipping through the air horizontally at a speed of 5 meters per second. This is the speed the skater imparted to the stone during the throw. It's crucial to remember that this is the horizontal velocity. In a real-world scenario, gravity would also be acting on the stone, causing it to fall vertically. However, for this particular problem, we're focusing solely on the horizontal motion and its effect on the skater due to the principle of conservation of momentum. The horizontal motion is what directly relates to the backward recoil of the skater because it's the horizontal force during the throw that causes the skater to move backward. So, this 5 m/s value is the crucial piece of information we need for the next step in our calculation. It’s a direct consequence of the force the skater applied and the time over which that force was applied, resulting in this specific speed for the stone.

Applying the Conservation of Momentum

Now, for the main event, guys: applying the conservation of momentum. This is where things get really interesting and where we can finally calculate that skater's speed. Remember our principle: in a closed system, the total momentum before an event equals the total momentum after. Initially, our skater and the stone are chilling together, presumably at rest. This means the total initial momentum ($p_{initial}$) of the system is zero. Think about it – if nothing is moving, nothing has momentum!

$$p_{initial} = 0$$

After the throw, we have two separate objects moving: the stone and the skater. The total final momentum ($p_{final}$) is the sum of the momentum of the stone ($p_{stone}$) and the momentum of the skater ($p_{skater}$).

$$p_{final} = p_{stone} + p_{skater}$$

According to the conservation of momentum, $p_{initial} = p_{final}$. Since $p_{initial} = 0$, we must have:

$$p_{stone} + p_{skater} = 0$$

This equation tells us that the momentum of the stone and the momentum of the skater must be equal in magnitude but opposite in direction. This is why the skater recoils – it's the universe's way of keeping the total momentum balanced!

We know that momentum ($p$) is calculated as mass ($m$) times velocity ($v$), so $p = mv$. We already calculated the stone's velocity ($v_{stone} = 5$ m/s) and we know its mass ($m_{stone} = 1$ kg). So, the stone's momentum is:

$$p_{stone} = m_{stone} \times v_{stone} = 1 \text{ kg} \times 5 \text{ m/s} = 5 \text{ kg}\cdot\text{m/s}$$

Now, let's bring in our skater. The skater's mass ($m_{skater}$) is 75 kg. We want to find the skater's final velocity ($v_{skater}$). So, the skater's momentum is:

$$p_{skater} = m_{skater} \times v_{skater} = 75 \text{ kg} \times v_{skater}$$

Substituting these into our conservation of momentum equation:

$$5 \text{ kg}\cdot\text{m/s} + (75 \text{ kg} \times v_{skater}) = 0$$

To find $v_{skater}$, we rearrange the equation:

$$75 \text{ kg} \times v_{skater} = -5 \text{ kg}\cdot\text{m/s}$$

$$v_{skater} = \frac{-5 \text{ kg}\cdot\text{m/s}}{75 \text{ kg}}$$

$$v_{skater} \approx -0.067 \text{ m/s}$$

The negative sign indicates that the skater moves in the opposite direction to the stone, which makes perfect sense! So, the skater acquires a speed of approximately 0.067 meters per second backward.