A model sailboat is slowly sailing west across a pond at 0.342 m/s. A gust of wind blowing at 28.7° south of west gives the sailboat a constant acceleration of magnitude 0.305 m/s2 during a time interval of 1.87 s. If the net force on the sailboat during the 1.87-s interval has magnitude 0.316 N, what is the sailboat's mass?
What is the new velocity of the boat after the 1.87 s gust of wind?
And what is the new direction of the boat after the 1.87 s gust of wind?
the only unbalanced force is the gust. That is , the steady wind driving the boat west provides no acceleration, so can be disregarded.
F = ma
.316 = m*.305
m = 1.036kg
during the gust, v is increased by a*t = 1.87*.305 = 0.570 = -.569i - .274j
final v = -.911i - .274j
|v| = .951
tanθ = -.274/-.911 = .300
θ = 16.7°, so W16.7°S
Why did the sailboat go to therapy?
Because it had trouble with its direction and needed some wind counseling! Now, let's solve this physics puzzle, shall we?
To find the mass of the sailboat, we can use Newton's second law, which states that force equals mass times acceleration: F = ma.
Given that the net force (F) is 0.316 N and the acceleration (a) is 0.305 m/s², we can rearrange the equation to solve for mass (m): m = F/a.
Plugging in the values, we get: m = 0.316 N / 0.305 m/s² = 1.035 kg.
So, the mass of the sailboat is approximately 1.035 kg.
Now, to find the new velocity of the boat after the 1.87 s gust of wind, we can use the equation of motion: v = u + at.
Here, "u" represents the initial velocity. Since the sailboat was sailing west at a constant velocity of 0.342 m/s, the initial velocity (u) is 0.342 m/s. The acceleration (a) is given as 0.305 m/s², and the time (t) is 1.87 s.
Let's calculate the new velocity: v = 0.342 m/s + (0.305 m/s²) * (1.87 s) = 0.342 m/s + 0.57035 m/s = 0.91235 m/s.
So, the new velocity of the sailboat after the gust of wind is approximately 0.912 m/s.
Lastly, let's determine the new direction of the boat after the 1.87 s gust of wind.
Since the initial direction was west, the gust of wind blowing at 28.7° south of west will change the direction. To find the new direction, we need to add the angle of the gust to the initial direction.
The new direction can be found using trigonometry. Subtract the angle of 28.7° from 180° (since south is 180°) to get the angle in the new direction.
180° - 28.7° = 151.3°.
Therefore, the new direction of the boat after the 1.87 s gust of wind is approximately 151.3° south of west.
And there you have it, the sailboat's mass, new velocity, and new direction!
Step 1: Determine the sailboat's mass
Given:
Acceleration (a) = 0.305 m/s^2
Net force (F) = 0.316 N
Using Newton's second law of motion, we can determine the mass (m) of the sailboat.
F = m * a
0.316 N = m * 0.305 m/s^2
m = 0.316 N / 0.305 m/s^2
m ≈ 1.036 kg
Therefore, the sailboat's mass is approximately 1.036 kg.
Step 2: Calculate the new velocity of the boat after the gust of wind
Given:
Initial velocity (u) = 0.342 m/s
Acceleration (a) = 0.305 m/s^2
Time (t) = 1.87 s
Using the equation of motion, v = u + at:
v = 0.342 m/s + (0.305 m/s^2 * 1.87 s)
v ≈ 0.342 m/s + 0.57035 m/s
v ≈ 0.91235 m/s
Therefore, the new velocity of the boat after the 1.87 s gust of wind is approximately 0.91235 m/s.
Step 3: Determine the new direction of the boat after the gust of wind
Given:
The gust of wind blows at 28.7° south of west.
Since the boat is initially sailing west, and the gust is from south of west, we need to find the resultant direction.
Using trigonometry, we can calculate the angle between the resulting direction and the west direction.
tan(θ) = (opposite/adjacent) = (Magnitude of south component / Magnitude of west component)
tan(θ) = sin(θ) / cos(θ) = (Magnitude of south component / Magnitude of west component)
tan(θ) = (0.305 m/s^2 * sin(28.7°)) / 0.342 m/s
θ ≈ tan^(-1)(0.305 m/s^2 * sin(28.7°) / 0.342 m/s)
θ ≈ tan^(-1)(0.1444 / 0.342)
θ ≈ tan^(-1)(0.4228)
θ ≈ 23.7°
Therefore, the new direction of the boat after the 1.87 s gust of wind is approximately 23.7° south of west.
To solve this problem, we can use Newton's second law of motion, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration.
Let's break down the given information. We know that the net force acting on the sailboat is 0.316 N and the acceleration caused by the gust of wind is 0.305 m/s^2. We need to find the mass of the sailboat.
Using Newton's second law equation, we can rearrange it to solve for mass (m):
Net force (F) = mass (m) * acceleration (a)
Plugging in the given values, we have:
0.316 N = m * 0.305 m/s^2
Now, we can solve for mass (m):
m = 0.316 N / 0.305 m/s^2
m ≈ 1.036 kg
So, the mass of the sailboat is approximately 1.036 kg.
Next, let's find the new velocity of the boat after the 1.87 s gust of wind. We can use the equation of motion:
Final velocity (v) = Initial velocity (u) + acceleration (a) * time (t)
The initial velocity of the boat is given as 0.342 m/s, and the acceleration caused by the gust of wind is 0.305 m/s^2. The time interval is 1.87 s.
Plugging in the given values, we have:
v = 0.342 m/s + 0.305 m/s^2 * 1.87 s
v ≈ 0.939 m/s
Therefore, the new velocity of the sailboat after the 1.87 s gust of wind is approximately 0.939 m/s.
Lastly, let's find the new direction of the boat after the gust of wind. The initial direction of the boat was west, and the gust of wind is blowing at 28.7° south of west.
To find the new direction, we can use trigonometry. We have a right triangle with the wind direction as the hypotenuse, the west direction as the adjacent side, and the south direction as the opposite side.
Using the tangent function, we can find the angle:
tan(θ) = opposite side / adjacent side
Plugging in the values, we have:
tan(θ) = sin(28.7°) / cos(28.7°)
θ ≈ 28.7° + 180°
θ ≈ 208.7°
Therefore, the new direction of the sailboat after the 1.87 s gust of wind is approximately 208.7° (southwest).