A boat that travels at 16 knots in calm water is sailing across a current of 3 knots on a river 250 m wide. The boat makes an angle of 35 degrees with the current heading into the current.
Find the resultant velocity of the boat. How far upstream is the boat when it reaches the other shore?
Do I need to use parametric equations to solve this? If so, how do I find the time? Thanks in advance.
Boat velocity component upstream = 16 cos 35 - 3
= 13.1 - 3
= 10.1 kn
Boat velocity component across stream = 16 sin 35 = 9.18 kn
resultant speed = sqrt(10.1^2+9.18^2) = 13.6 kn
tangent of resultant angle to upstream = 9.18/10.1 = .909
so angle to upstream = tan^-1 .909 = 42.3 deg
Now you do not have to know time T because
distance upstream = 10.1 T
distance across = 9.18 T
distance upstream/distance across =10.1/9.18 (The T cancels
so
10.1/9.18 = distance upstream/250 meters
so
distance upstream = 250 * 10.1/9.18 = 275 meters
That avoids having to convert nautical miles to meters. I suppose in a sense my use of T implies parametric :)
To solve this problem, you don't necessarily need to use parametric equations. You can use vector addition to find the resultant velocity of the boat.
Let's break down the velocities into components. The boat's velocity in calm water can be divided into horizontal and vertical components:
Vb_horizontal = 16 knots * cos(35°) = 13.07 knots
Vb_vertical = 16 knots * sin(35°) = 9.23 knots
The current's velocity can also be divided into horizontal and vertical components:
Vc_horizontal = 3 knots
Vc_vertical = 0 knots (since the current flows horizontally)
Now, let's calculate the resultant velocity of the boat. Since the boat is heading into the current, we can subtract the current's velocity components from the boat's velocity components:
V_r_horizontal = Vb_horizontal - Vc_horizontal = 13.07 knots - 3 knots = 10.07 knots
V_r_vertical = Vb_vertical - Vc_vertical = 9.23 knots - 0 knots = 9.23 knots
Using the Pythagorean theorem, we can find the magnitude of the resultant velocity:
V_r = sqrt(V_r_horizontal^2 + V_r_vertical^2)
= sqrt((10.07 knots)^2 + (9.23 knots)^2)
≈ 14.0 knots
Now, to find the time it takes for the boat to cross the river, we can use the formula:
Time = Distance / Velocity
Since the boat is traveling directly across the 250 m wide river, the distance traveled is 250 m. Therefore:
Time = 250 m / (14.0 knots * 0.5144 m/s per knot)
≈ 11.96 seconds
Finally, we can calculate how far upstream the boat traveled by multiplying the horizontal component of the resultant velocity by the time:
Distance upstream = V_r_horizontal * Time
= 10.07 knots * 0.5144 m/s per knot * 11.96 seconds
≈ 62.15 meters
So, the resultant velocity of the boat is approximately 14.0 knots, and it travels approximately 62.15 meters upstream when it reaches the other shore.
To find the resultant velocity of the boat, we can consider the boat's velocity relative to the ground and the velocity of the current.
Given:
- Boat's velocity in calm water: 16 knots
- Velocity of the current: 3 knots
- Width of the river: 250 m
- Angle between the boat's heading and the current: 35 degrees
To find the resultant velocity of the boat, we can use vector addition. Let's break down the velocities into their components:
- Boat's velocity in calm water: The horizontal component is 16 knots, and the vertical component is 0 knots since the current doesn't affect the boat's vertical motion.
- Velocity of the current: The horizontal component is 0 knots since the current doesn't affect the boat's horizontal motion, and the vertical component is 3 knots.
Now, let's add the components to find the resultant velocity:
Horizontal component of resultant velocity = Horizontal component of boat's velocity + Horizontal component of current's velocity
= 16 knots + 0 knots
= 16 knots
Vertical component of resultant velocity = Vertical component of boat's velocity + Vertical component of current's velocity
= 0 knots + 3 knots
= 3 knots
Now, we have the horizontal and vertical components of the resultant velocity. To find the magnitude and direction of the resultant velocity, we can use the Pythagorean theorem and inverse tangent:
Magnitude of resultant velocity = sqrt((Horizontal component)^2 + (Vertical component)^2)
= sqrt((16 knots)^2 + (3 knots)^2)
= sqrt(256 knots^2 + 9 knots^2)
= sqrt(265 knots^2)
≈ 16.28 knots
Angle of resultant velocity = arctan(Vertical component / Horizontal component)
= arctan(3 knots / 16 knots)
≈ 10.94 degrees
Therefore, the resultant velocity of the boat is approximately 16.28 knots at an angle of approximately 10.94 degrees from the horizontal.
To find how far upstream the boat is when it reaches the other shore, we need to find the time it takes to cross the river. Since the boat is traveling at an angle, we can use parametric equations to solve this.
Let's denote the time it takes for the boat to cross the river as "t". The horizontal distance covered by the boat can be calculated as the horizontal velocity multiplied by time:
Distance = Horizontal component of resultant velocity * time
= 16 knots * t
Since the width of the river is given as 250 m, we can equate the distance with the width of the river to find the time:
16 knots * t = 250 m
To find the time t, we need to convert the knots to meters per second. 1 knot is approximately equal to 0.5144 m/s. Therefore, 16 knots is equal to 16 * 0.5144 m/s.
16 * 0.5144 m/s * t = 250 m
Solving for t:
t ≈ 9.735 seconds
Therefore, it takes approximately 9.735 seconds for the boat to cross the river.
To find how far upstream the boat is when it reaches the other shore, we can use the formula:
Distance upstream = Vertical component of resultant velocity * time
Distance upstream = 3 knots * t (as time is already in seconds)
Again, we need to convert the knots to meters per second:
3 * 0.5144 m/s * t ≈ 14.557 m
Therefore, the boat is approximately 14.557 meters upstream when it reaches the other shore.