A motor boat whose speed is 2.6 m/s in still water, needs to point upstream at an angle of 28.5 in order to travel straight across the river. Calculate the speed of the current and the resultant velocity of the boat across the river.
current speed is 2.6 sin 28.5°
boat velocity across is 2.6 cos 28.5°
To solve this problem, we can break the velocity of the boat into two components: one along the direction of the river, and one perpendicular to the river.
Let's assume the speed of the current is "c" m/s.
The component of the boat's velocity along the direction of the river is given by:
Velocity along the river = Speed of the boat * cos(angle)
= 2.6 m/s * cos(28.5°)
The component of the boat's velocity perpendicular to the river (resultant velocity across the river) is given by:
Velocity across the river = Speed of the boat * sin(angle)
= 2.6 m/s * sin(28.5°)
Since the boat needs to travel straight across the river, the resultant velocity across the river must be zero. Therefore:
Velocity across the river = Velocity of the current
So we have:
2.6 m/s * sin(28.5°) = c,
where c is the speed of the current.
Now we can solve for c:
c = 2.6 m/s * sin(28.5°)
c ≈ 1.43 m/s
Therefore, the speed of the current is approximately 1.43 m/s.
To calculate the resultant velocity of the boat across the river, we substitute c back into the equation for Velocity across the river:
Velocity across the river = 2.6 m/s * sin(28.5°)
Velocity across the river ≈ 2.6 m/s * 0.4848
Velocity across the river ≈ 1.26 m/s
Therefore, the resultant velocity of the boat across the river is approximately 1.26 m/s.
To calculate the speed of the current and the resultant velocity of the boat across the river, we can use the concept of vector addition.
Let's break down the motion of the boat into two components: the velocity of the boat in still water (Vb) and the velocity of the current (Vc).
Given:
Speed of the boat in still water (Vb) = 2.6 m/s
Angle between the boat's direction and upstream direction = 28.5°
First, we need to find the velocity of the boat with respect to the ground (resultant velocity). This can be calculated using trigonometry.
The vertical component of the boat's velocity is Vbv = Vb * sin(angle).
Vbv = 2.6 m/s * sin(28.5°)
The horizontal component of the boat's velocity is Vbh = Vb * cos(angle).
Vbh = 2.6 m/s * cos(28.5°)
Now, let's consider the current. Since the boat needs to point upstream to travel straight across the river, the current must be coming from the right side of the river towards the boat. Therefore, the horizontal component of the current's velocity is negative.
Let's assume the speed of the current is Vc in m/s.
So, the vertical component of the current's velocity is Vcv = 0 m/s (since the current flows horizontally).
The horizontal component of the current's velocity is Vch = -Vc (negative because it's in the opposite direction).
To find the resultant velocity of the boat across the river, we need to add the horizontal components of the boat's velocity and the current's velocity.
Resultant velocity: Vrh = Vbh + Vch
To find the speed of the current, we can use the concept of vector addition. Since the boat is traveling at an angle of 28.5° with respect to the river's flow direction, the resultant velocity (Vr) can be calculated as:
Vr = sqrt(Vrh^2 + Vrv^2)
Where Vr represents the speed of the resultant velocity of the boat (across the river) and Vrv represents the vertical component of the resultant velocity (which is zero).
Now, we can calculate the values.
Step 1: Calculate the vertical component of the boat's velocity (Vbv).
Vbv = 2.6 m/s * sin(28.5°)
Vbv ≈ 1.35 m/s
Step 2: Calculate the horizontal component of the boat's velocity (Vbh).
Vbh = 2.6 m/s * cos(28.5°)
Vbh ≈ 2.33 m/s
Step 3: Calculate the horizontal component of the current's velocity (Vch).
Vch = -Vc
Step 4: Calculate the resultant velocity of the boat (Vrh).
Vrh = Vbh + Vch
Vrh = 2.33 m/s + (-Vc)
Step 5: Calculate the speed of the resultant velocity of the boat across the river (Vr).
Vr = sqrt(Vrh^2 + Vrv^2)
Vr = sqrt((2.33 m/s + (-Vc))^2 + (0 m/s)^2)
Now, you can solve the equations to find the value of Vc and Vr.