On spring break in Puerto Vallarta you are feeling adventurous and decide to try ocean kayaking. You ate too much and now your mass is 85 kg, the mass of the kayak is 150 kg. You start from the beach directly in front of your hotel going 20 m/s due west, with an ocean current going 10 m/s due south. After 30 min where will you be in relation to your hotel?
Well, it sounds like you've got quite the situation there! Let's see where you end up after all that kayaking and eating.
First, let's break down your velocity components. You're moving west at 20 m/s, and the ocean current is pushing you south at 10 m/s. So, your resulting velocity will be a combination of these two vectors.
Since we're talking about velocities, we need to consider the change in position over time. In this case, you've been kayaking for 30 minutes, which is equal to 0.5 hours.
Now, let's calculate the westward distance you covered. In half an hour, you've traveled 20 m/s * 0.5 h = 10 meters west.
Next, let's calculate the southward distance due to the current. Similarly, the distance you covered is 10 m/s * 0.5 h = 5 meters south.
Brace yourself for the punchline: combining these two distances, you'll end up 10 meters west and 5 meters south from your starting point. So, after all that paddling and eating, you'll find yourself in the ocean, 10 meters west and 5 meters south of your hotel. Just remember to bring a snorkel and some sunscreen along!
To determine your final position after 30 minutes of ocean kayaking, we need to consider the velocities and the time elapsed.
Given information:
- Your mass: 85 kg
- Kayak mass: 150 kg
- Initial velocity (your velocity): 20 m/s due west
- Ocean current velocity: 10 m/s due south
- Time elapsed: 30 minutes (which is equivalent to 0.5 hours)
Let's break it down step-by-step:
Step 1: Convert time to seconds
Since the given time is in minutes, we need to convert it to seconds:
30 minutes = 30 × 60 = 1800 seconds
Step 2: Calculate net force
The net force can be calculated using the formula:
Net Force = (Mass of the kayak + Your mass) × Acceleration
The acceleration can be calculated as:
Acceleration = (Your Velocity + Current Velocity) / Time
Acceleration = (20 m/s + 10 m/s) / 1800 s
Acceleration = 30 m/s / 1800 s
Acceleration ≈ 0.0167 m/s²
Net Force = (150 kg + 85 kg) × 0.0167 m/s²
Net Force = 235 kg × 0.0167 m/s²
Net Force ≈ 3.9205 N
Step 3: Calculate displacement
To calculate your displacement, we can use the equation:
Displacement = Initial Velocity × Time + (1/2) × Acceleration × Time²
Since our initial velocity is in meters per second (m/s), and our time is in seconds (s):
Displacement = (20 m/s × 1800 s) + (1/2) × 0.0167 m/s² × (1800 s)²
Displacement = 36,000 m + (1/2) × 0.0167 m/s² × 3,240,000 s²
Displacement ≈ 36,000 m + 27,090 m
Displacement ≈ 63,090 m
Step 4: Determine direction
Given that you started from the beach directly in front of your hotel, moving west, and the ocean current is pulling you south, the displacement will be a vector addition of your initial velocity and the current velocity.
Using Pythagorean's theorem, let's calculate the final displacement's magnitude and direction:
Magnitude of displacement = √(Displacement west)² + (Displacement south)²
Magnitude of displacement = √(63,090 m)² + (1800 m)²
Magnitude of displacement ≈ √3,976,848,100 m² + 3,240,000 m²
Magnitude of displacement ≈ √4,009,088,100 m²
Magnitude of displacement ≈ 63,331 m
To determine the direction, we can use inverse tangent (arctan) to find the angle:
Direction = arctan(Displacement south / Displacement west)
Direction = arctan(1800 m / 63,090 m)
Direction ≈ arctan(0.0285)
Direction ≈ 1.63° south of west
Thus, after 30 minutes of ocean kayaking, you will be approximately 63,331 meters away from your hotel, in a direction 1.63° south of west.
To determine your final position after 30 min of ocean kayaking, we need to consider your initial velocity, the ocean current, and the time elapsed.
1. Calculate the total initial velocity:
Your initial velocity is given as 20 m/s due west. However, since it's a kayak, we need to take your mass into account. The net (effective) force acting on you will be the difference between the force you generate by paddling and the force of the ocean current pushing against you. The force is equal to mass times acceleration (F = ma). The acceleration is the change in velocity per unit time, which we denote as Δv/Δt. For example, if your velocity changes by 20 m/s in 30 min, then the acceleration would be (20 m/s) / (30 min).
2. Calculate the acceleration using masses:
Start by calculating the acceleration caused by your mass using Newton's second law: F = ma. Rearranging the equation, you get a = F/m. You need to determine the force you generate by paddling, which is equal to your mass times the acceleration caused by your movement. With the given information, calculate this force.
3. Calculate the acceleration due to the ocean current:
The force of the ocean current pushing against you is given by the mass of the kayak times the acceleration caused by the current. Use the given velocity and time duration to calculate the acceleration caused by the current.
4. Calculate the net acceleration:
The net acceleration is the difference between the acceleration caused by your movement and the acceleration caused by the ocean current. Calculate this net acceleration.
5. Determine the displacement:
Now, the displacement can be found using the equation: Δx = v0t + (1/2)aΔt^2, where:
- Δx is the displacement (distance and direction)
- v0 is the initial velocity (20 m/s due west)
- t is the time elapsed (30 min, which needs to be converted to seconds)
- a is the net acceleration
6. Find your final position:
Add the displacement to your starting position (the beach directly in front of your hotel) to determine where you will be in relation to your hotel.
Note: Ensure to convert units appropriately (e.g., convert 30 min to seconds) and apply the appropriate sign conventions.
Following these steps, you will be able to determine your final position in relation to your hotel after 30 min of ocean kayaking.