A girl swims across a river. When she swims in still water, she swims at 1.25 m/s, the river flows parallel to its banks at v m/s. The girl aims to swim upstream at angle θ degrees parallel to the river bank so that her resultant velocity of 1 m/s is along AB, perpendicular to the river bank.
Find V and θ
sin θ = 1/1.25
tan θ = sin θ/cos θ = 1/v
To find the value of V and θ, we can use vector addition to find the resultant velocity of the girl as she swims across the river.
Let's break down the problem into components:
- The girl's velocity in still water: 1.25 m/s
- The river's velocity: v m/s
- The angle θ between the girl's desired upstream swimming direction and the river bank
- The resultant velocity of the girl: 1 m/s along AB (perpendicular to the river bank)
Now, let's analyze the situation using vector addition. We need to find the values of V and θ such that the resultant velocity is 1 m/s along AB.
First, let's resolve the girl's velocity into its horizontal and vertical components. The vertical component of the girl's velocity will be 1.25sinθ m/s since she is aiming upstream at angle θ. The horizontal component will be 1.25cosθ m/s.
Next, let's consider the river's velocity. Since the river flows parallel to its banks, its velocity is entirely horizontal. Therefore, the horizontal component of the river's velocity is v m/s, and the vertical component is 0 m/s.
Now, let's add the horizontal and vertical components of the girl's velocity and the river's velocity to find the resultant velocity.
Horizontal Component:
Girl's horizontal velocity component: 1.25cosθ m/s
River's horizontal velocity component: v m/s
Vertical Component:
Girl's vertical velocity component: 1.25sinθ m/s
River's vertical velocity component: 0 m/s
Resultant Horizontal Velocity: 1.25cosθ + v m/s
Resultant Vertical Velocity: 1.25sinθ m/s
We want the resultant velocity to be 1 m/s along AB, which means the vertical component should be 0 m/s.
Therefore, we set the resultant vertical velocity equal to 0:
1.25sinθ = 0
Since sinθ = 0 when θ = 0 or θ = 180 degrees, we have two possibilities for the angle θ.
Case 1:
When θ = 0 degrees,
1.25sin0 = 0
0 = 0
Case 2:
When θ = 180 degrees,
1.25sin180 = 0
0 = 0
So, both Case 1 and Case 2 satisfy the condition for the resultant vertical velocity to be 0 m/s.
Now, let's find the value of V (the resultant horizontal velocity) when θ = 0 degrees.
Resultant Horizontal Velocity: 1.25cosθ + v
When θ = 0 degrees,
Resultant Horizontal Velocity: 1.25cos0 + v
Resultant Horizontal Velocity: 1.25 + v m/s
So, when θ = 0 degrees, the resultant horizontal velocity is 1.25 + v m/s.
Therefore, the values of V and θ are:
V = 1.25 + v m/s
θ = 0 degrees
Note: It's important to verify the units of the given quantities and ensure they are consistent throughout the calculations.