Relative Motion in Two Dimensions — Question 6

Question 6

Apparent rain direction is a relative velocity

Use x forward along a car’s eastward motion and y upward. Rain falls vertically at 8 m/s relative to ground, while the car moves east at 6 m/s. Neglect airflow changes caused by the car.

1. Find the rain velocity and speed relative to the car.

2. Find the angle of the apparent falling direction from vertically downward and identify its horizontal sense.

3. From which tilted upward direction does the rain appear to come? Explain the distinction between travel direction and arrival direction.

Question 6 - Solution

Apparent rain direction is a relative velocity

(a) Subtract the observer velocity.

$$\boxed{\vec v_{R/C}=(0,-8)-(6,0)=(-6,-8)\,\mathrm{m/s}},\quad v_{R/C}=10\,\mathrm{m/s}$$ $\vec v_{R/C}=\vec v_{R/G}-\vec v_{C/G}=(0,-8)-(6,0)=(-6,-8)$ m/s. Apparent speed is $\sqrt{36+64}=10$ m/s. The car adds a westward relative component even though the ground rain motion is vertical.

(b) Use the downward reference axis.

$$\tan\theta=\frac{|v_x|}{|v_y|}=\frac68\ \Rightarrow\ \theta=36.870^\circ\quad\text{rearward from downward vertical}$$ The angle is $\tan^{-1}(6/8)\approx36.870^\circ$ from downward vertical toward the rear of the car. This describes the direction in which droplets move past the observer.

(c) Reverse the velocity to trace droplets backward.

$$\hat n_{\mathrm{arrival}}=-\frac{\vec v_{R/C}}{|\vec v_{R/C}|}=\frac{(6,8)}{10}=(.6,.8)$$ The rain appears to come from the direction $(6,8)$, upward and forward, tilted $36.870^\circ$ from upward vertical toward the front. The direction a moving object comes from is opposite its velocity direction. Confusing these two directions reverses the apparent slant.