Horizontal veolocity given vertical velocity and angle

[IAmSparticus]

Homework Statement

A person looking out the window of a stationary train notices that raindrops are falling vertically down at a speed of 8.7 m/s relative to the ground. When the train moves at a constant velocity, the raindrops make an angle of θ = 67° when they move past the window as the drawing shows. How fast is the train moving?

Homework Equations

Vx = V0x +axt

The Attempt at a Solution

3.4 and 22.3 m/s are both incorrect.

Couldn't figure out how to do it since clearly you don't just use the Pythagorean Theorem... walk me through it?

 

[tiny-tim]

Hi IAmSparticus!

You'd only use Pythagoras if you wanted the length of the third side.

But you only want the angle … so use cos or sin or tan.

 

[tomcue]

I would first start by analyzing the given information and identifying the relevant variables and equations that can be used to solve for the unknown variable, which in this case is the horizontal velocity of the train.

From the given information, we know that the raindrops are falling vertically at a speed of 8.7 m/s relative to the ground. This means that the vertical component of the raindrop's velocity is 8.7 m/s. We also know that when the train is moving, the raindrops make an angle of θ = 67° as they pass by the window.

To solve for the horizontal velocity, we can use the equation Vx = V0x +axt, where Vx is the horizontal velocity, V0x is the initial horizontal velocity (which is 0 since the train starts from rest), a is the acceleration (which is also 0 since the train is moving at a constant velocity), and t is the time.

We can use trigonometry to find the value of V0x. Since we know that the raindrops make an angle of θ = 67° with the horizontal, we can use the trigonometric function cosine to find the value of V0x. The equation for this would be V0x = V0cosθ, where V0 is the initial velocity (in this case, the vertical velocity of the raindrops).

Substituting the known values, we get V0x = 8.7 m/s * cos(67°) = 3.4 m/s.

Now, we can plug this value into the original equation Vx = V0x +axt to solve for the horizontal velocity. Since a = 0, the equation simplifies to Vx = V0x, which is equal to 3.4 m/s.

Therefore, the train is moving at a speed of 3.4 m/s. This is the correct answer, as both 3.4 and 22.3 m/s were incorrect because they did not take into account the angle at which the raindrops were falling.