A man observes snow falling vertically when he is at rest, but when he runs through the falling snow at a speed of 4.70 m/s, it appears to be falling at an angle of 24.0° relative to the vertical. Find the speed of the snow relative to the earth.
(4.70 m/s)/Vsnow = tan24
Solve for Vsnow (the vertical velocity component)
To find the speed of the snow relative to the earth, we need to use vector addition.
Let's define some variables:
- v_snow is the speed of the snow relative to the man.
- v_man is the velocity of the man relative to the earth, which is 4.70 m/s horizontally.
- v_snow/earth is the speed of the snow relative to the earth, which we want to find.
- θ is the angle that the snow appears to fall at relative to the vertical.
The velocity of the snow relative to the man can be split into horizontal and vertical components. The horizontal component is the same as the man's velocity, which is v_man = 4.70 m/s. The vertical component can be found using trigonometry:
v_snow_y = v_snow * sin(θ)
Since the man observes the snow falling vertically when he is at rest, we know that v_snow_y = 0. Therefore, we can write:
0 = v_snow * sin(θ)
Solving for v_snow, we get:
v_snow = 0 / sin(θ) = 0
Now, we can solve for v_snow/earth using vector addition.
The vector sum of v_snow and v_man gives us v_snow/earth:
v_snow/earth = sqrt((v_snow)^2 + (v_man)^2)
Since v_snow is 0, the formula simplifies to:
v_snow/earth = sqrt(0^2 + (4.70 m/s)^2)
v_snow/earth = sqrt(0 + 22.09 m^2/s^2)
v_snow/earth = sqrt(22.09 m^2/s^2)
Calculating the square root, we find:
v_snow/earth ≈ 4.70 m/s
Therefore, the speed of the snow relative to the earth is approximately 4.70 m/s.