You're sailboarding at 6.1m/s when a wind gust hits, lasting 6.8s accelerating your board at 0.49m/s2 at a 35∘ to your original direction.
1. Find the magnitude of your displacement during the gust.
Express your answer using two significant figures.
2. Find direction of your displacement during the gust.
Express your answer using two significant figures.
1. D = Vo*t + 0.5a*t^2 =
6.1*6.8 * 0.5*0.49[35o]*6.8^2 = 41.48 +
11.33[35o] = 41.48 + 9.28+6.5.0i =
50.76 + 6.5i
D^2 = 50.76^2 + 6.5^2 = 2618.3
D = 51 m.
2. Tan A = Y/X = 6.5/50.76 = 0.12805
A = 7.3o = Direction.
1. Well, isn't that quite a gusty situation you've got there! To find the magnitude of your displacement during the gust, we can use the equation:
Displacement = Initial Velocity × Time + 0.5 × Acceleration × Time^2
Plugging in the given values, we get:
Displacement = (6.1 m/s × 6.8 s) + 0.5 × (0.49 m/s^2) × (6.8 s)^2
Calculating that out, we find the magnitude of your displacement during the gust to be approximately 51.45 meters.
2. Now, to determine the direction of your displacement during this windy adventure! Since the gust is at an angle of 35 degrees to your original direction, we can say that your displacement is also at that same angle. So, the direction of your displacement during the gust is approximately 35 degrees from your original direction.
Keep sailboarding and may your gusts always be gusty!
To find the magnitude of your displacement during the gust, we can use the formula for displacement:
Displacement (d) = initial velocity (v0) * time (t) + 0.5 * acceleration (a) * time squared (t^2)
Given:
Initial velocity (v0) = 6.1 m/s
Acceleration (a) = 0.49 m/s^2
Time (t) = 6.8 s
Using the given values, we can calculate the displacement as follows:
Displacement (d) = (6.1 m/s) * (6.8 s) + 0.5 * (0.49 m/s^2) * (6.8 s)^2
Calculating this expression, we find:
Displacement (d) = 41.48 m
Therefore, the magnitude of your displacement during the gust is approximately 41.48 meters (rounded to two significant figures).
Now, let's calculate the direction of your displacement during the gust.
Given:
Angle of acceleration (θ) = 35 degrees
The direction of displacement during the gust will be at a 35-degree angle to your original direction.
Therefore, the direction of your displacement during the gust is approximately 35 degrees (rounded to two significant figures) relative to your original direction.
To find the magnitude of your displacement during the gust, we can use the equations of motion.
1. First, let's find the initial velocity component in the direction of the gust. We know that your sailboarding velocity is 6.1 m/s and the gust is at an angle of 35° to your original direction. Using trigonometry, we can find the initial velocity component in the direction of the gust:
Initial velocity component = 6.1 m/s * cos(35°)
2. Next, let's find the final velocity component in the direction of the gust. The wind gust accelerates your board at a rate of 0.49 m/s² for 6.8 seconds. Using the equation:
Final velocity component = Initial velocity component + (acceleration * time)
3. Now, we can find the displacement in the direction of the gust using the equation:
Displacement = (initial velocity component + final velocity component) / 2 * time
Now let's calculate the values:
Initial velocity component = 6.1 m/s * cos(35°) ≈ 4.99 m/s
Final velocity component = 4.99 m/s + (0.49 m/s² * 6.8 s) ≈ 8.32 m/s
Displacement = (4.99 m/s + 8.32 m/s) / 2 * 6.8 s ≈ 30.5 m
Therefore, the magnitude of your displacement during the gust is approximately 30.5 meters, rounded to two significant figures.
To find the direction of your displacement during the gust, we can use the following equation:
Direction = arctan(final velocity component / initial velocity component)
Substituting the values:
Direction = arctan(8.32 m/s / 4.99 m/s) ≈ 60.6°
Therefore, the direction of your displacement during the gust is approximately 60.6°, rounded to two significant figures.