You are driving south on a highway at 25 m/s (approximately 55 mi/h) in a snow storm. When you last stopped, you noticed that the snow was coming down vertically, but it is passing the windows of the moving car at an angle of 30 degrees to the horizontal. Estimate the speed of the snowflakes relative to the car and relative to the ground
Well, well, well, we have a snow-storm driving situation! Let me put on my snow goggles and calculate this for you.
If the snowflakes are passing the windows at an angle of 30 degrees to the horizontal, we can break it down into vertical and horizontal components. The vertical component of the snowflakes' speed is perpendicular to the direction of your motion, so it doesn't affect the car's speed.
Now, the horizontal speed of the snowflakes relative to the car is what we're after. Since you're driving at a steady speed of 25 m/s southward, and the snowflakes are passing by at a 30-degree angle, we can use some trigonometry to find the horizontal component.
The horizontal speed of the snowflakes relative to the car can be found using the equation: Horizontal speed = Snowflake speed × cos(angle).
So, the horizontal speed of the snowflakes relative to the car is approximately 25 m/s × cos(30°).
As for the speed of the snowflakes relative to the ground, we just need to add the speed of the car to the horizontal speed of the snowflakes relative to the car.
Therefore, the speed of the snowflakes relative to the ground is approximately 25 m/s + (25 m/s × cos(30°)).
Now, if you want me to convert these speeds to clown units, you'll have to book me for a comedy show! Otherwise, I hope these calculations help you understand the relative speeds of the snowflakes. Drive safe in that snowstorm! 🚗❄️
To estimate the speed of the snowflakes relative to the car and relative to the ground, we can break down the velocities into their horizontal and vertical components.
Step 1: Calculate the horizontal component of the car's velocity
Given that the car is driving south, we assume the horizontal component of its velocity is zero. Since the snowflakes are passing the windows at an angle of 30 degrees to the horizontal, we can deduce that their horizontal velocity relative to the car is also zero.
Step 2: Calculate the vertical component of the car's velocity
The car's vertical velocity is given by the rate at which the snowflakes are passing the windows. The snowflakes' vertical velocity relative to the car is therefore 25 m/s (given in the problem).
Step 3: Calculate the speed of the snowflakes relative to the car
Using the Pythagorean theorem, we can calculate the speed of the snowflakes relative to the car:
Speed^2 = (Vertical Component)^2 + (Horizontal Component)^2
Speed^2 = (25 m/s)^2 + (0 m/s)^2
Speed = 25 m/s
The speed of the snowflakes relative to the car is 25 m/s.
Step 4: Calculate the speed of the snowflakes relative to the ground
To calculate the speed of the snowflakes relative to the ground, we need to consider the motion of the car as well as the snowflakes. Since the car is moving south at 25 m/s and the snowflakes are passing the windows at an angle of 30 degrees to the horizontal, we can use trigonometry to find the horizontal component of the snowflakes' velocity relative to the ground.
Horizontal Component = Speed * cos(Angle)
Horizontal Component = 25 m/s * cos(30 degrees)
Using the cosine of 30 degrees, which is √3/2,
Horizontal Component = 25 m/s * (√3/2)
Horizontal Component ≈ 21.65 m/s (rounded to two decimal places)
The speed of the snowflakes relative to the ground is approximately 21.65 m/s (rounded to two decimal places).
To estimate the speed of the snowflakes relative to the car and relative to the ground, we need to consider the motion of the car and the angle at which the snowflakes pass the car's window.
Let's break down the problem step by step:
Step 1: Convert the car's speed from m/s to km/h for simplicity.
25 m/s is equivalent to (25 * 3600) / 1000 = 90 km/h (rounded to the nearest whole number).
Step 2: Find the vertical component of the snowflakes' velocity relative to the car.
Since the snow was coming down vertically when you last stopped, we can assume that the vertical component of the snowflakes' velocity relative to the car is 0 m/s.
Step 3: Find the horizontal component of the snowflakes' velocity relative to the car.
Given that the snowflakes are passing the car's window at an angle of 30 degrees to the horizontal, we can use trigonometry to find the horizontal component of their velocity.
Using the cosine function:
cos(30 degrees) = horizontal velocity / relative velocity to the car
cos(30 degrees) = horizontal velocity / relative velocity
The horizontal velocity is the velocity of the snowflakes relative to the ground, which is what we are trying to find. Let's call it V_ground.
cos(30 degrees) = V_car / V_ground
V_ground = V_car / cos(30 degrees)
V_ground = 90 km/h / cos(30 degrees)
Using a calculator, the value of cos(30 degrees) is approximately 0.866.
V_ground = 90 km/h / 0.866
V_ground ≈ 103.92 km/h
Therefore, the speed of the snowflakes relative to the ground is approximately 103.92 km/h.
Step 4: Find the speed of the snowflakes relative to the car.
Since the vertical component of their velocity is 0 m/s and the horizontal component is the speed of the car, we can use the Pythagorean theorem to find the magnitude of their relative velocity to the car.
The magnitude of the relative velocity to the car is given by:
Relative velocity to the car = √(horizontal velocity^2 + vertical velocity^2)
Since the vertical velocity is 0 m/s:
Relative velocity to the car = √(V_car^2 + 0^2)
Relative velocity to the car = V_car
Therefore, the speed of the snowflakes relative to the car is approximately 90 km/h (same as the car's speed).
To summarize:
- The speed of the snowflakes relative to the ground is approximately 103.92 km/h.
- The speed of the snowflakes relative to the car is approximately 90 km/h.