Summary Relative Velocity Applications: Rain and Umbrella – Conceptual & Numerical Examples

Relative Velocity Applications: Rain and Umbrella Problems – Solved Examples


Example ① (Basic Concept):

Rain falls vertically with a constant velocity relative to the ground (𝑣⃗𝐴/𝐶 ),
and a person moves horizontally with a constant velocity relative to the A A/C


ground (𝑣⃗𝐵/𝐶 ). Answer the following:
a) Determine the velocity of the rain relative to the person (𝑣⃗𝐴/𝐵 ).
b) In which direction should the person hold the umbrella to avoid getting wet? B
B/C

C
Answer:
a)
C/B
𝑣⃗𝐴/𝐵 = 𝑣⃗𝐴/𝐶 + 𝑣⃗𝐶/𝐵
2 2
= √𝑣𝐴∕𝐶 + 𝑣𝐶∕𝐵
A/C


A/B




important rule: (changing the reference frame reverses the direction). Therefore, the direction of (𝑣⃗𝐶/𝐵 ) is
opposite to the direction of (𝑣⃗𝐵/𝐶 ).


b) The direction of the rain’s velocity relative to the person is inclined, A/B


tilted toward the back and downward. Therefore, the umbrella should be
held tilted forward, opposite to the relative velocity of the rain with
A
respect to the person (𝑣⃗𝐴/𝐵 ).
B

C



Example ② (With Given Velocities)

Rain falls vertically downward with a velocity of (4𝑚/𝑠) relative to the ground. A person moves horizontally
with a velocity of (3𝑚/𝑠) relative to the ground:

a) Determine the velocity of the rain relative to the person.
b) Find the angle at which the umbrella should be held.

Given: 𝑣⃗𝐴/𝐶 = −4𝑚/𝑠 , 𝑣⃗𝐵/𝐶 = +3𝑚/𝑠
Required: 𝑣𝐴/𝐵 , 𝜃
Solution:
First, you must remember that:

𝑣⃗𝐶/𝐵 = −𝑣⃗𝐵/𝐶 = −3𝑚/𝑠 (backwards)