Summary Relative Velocity Applications: Boat Crossing River (2D Motion) – Solved Problems

Relative Velocity Applications: Boat Crossing River (2D Motion) – Solved Problems


Example ① (Basic 2D Motion):

A boat (𝐴) moves in a river (𝐵) with a velocity of (3𝑚/𝑠) relative to the river. The river flows with a velocity
of (4𝑚/𝑠) relative to the ground (𝐶):
a) Find the magnitude of the boat’s velocity relative to the ground if the boat moves perpendicular to the
river flow.
b) Find the direction of the boat’s velocity relative to the ground.
B/C =4m/s



Given: 𝑣⃗𝐴/𝐵 = +3𝑚/𝑠 , 𝑣⃗𝐵/𝐶 = +4𝑚/𝑠
Required: 𝑣𝐴/𝐶 , 𝜃
Solution:
𝑣⃗𝐴/𝐶 = 𝑣⃗𝐴/𝐵 + 𝑣⃗𝐵/𝐶

2 2
A/B =3m/s
= √𝑣𝐴∕𝐵 + 𝑣𝐵∕𝐶
B/C



= √(3)2 + (4)2

= 5𝑚/𝑠
𝑣𝐵/𝐶 4 A/B
𝑡𝑎𝑛 𝜃 = =
𝑣𝐴/𝐵 3

→ 𝜃 ≈ 53° (𝑛𝑜𝑟𝑡ℎ 𝑜𝑓 𝑒𝑎𝑠𝑡)




Example ② (With Drift):

A river (𝐵) is (120𝑚) wide and flows at (4𝑚/𝑠) relative to the ground (𝐶). A boat (𝐴) moves at (5𝑚/𝑠)
relative to the river, directed straight toward the opposite bank:
a) Calculate the time taken to cross the river.
b) Calculate the distance the boat drifts downstream while crossing.
d


Given: 𝑣⃗𝐴/𝐵 = +5𝑚/𝑠 , 𝑣⃗𝐵/𝐶 = +4𝑚/𝑠 A/B =5m/s
Required: t , 𝑑
Solution: w=120m
The idea of the problem is that the horizontal velocity B/C =4m/s
is related to the horizontal displacement, and the vertical
velocity is related to the vertical displacement.