in a race, a boat travels a distance of 220m [E40N] and then rounds a buoy and travels a distance of 360m [N30W] to the finish line. the whole trip takes 22 seconds. determine the displacement and average velocity of the boat.
To determine the displacement of the boat, we can calculate the sum of the displacement vectors for each leg of the race.
For the first leg, the boat travels 220m towards the east while heading 40 degrees north of the east direction. To find the horizontal component, we can use the cosine function:
Horizontal component = 220m * cos(40°) ≈ 168.69m towards the east
To find the vertical component, we can use the sine function:
Vertical component = 220m * sin(40°) ≈ 141.46m towards the north
For the second leg, the boat travels 360m towards the north while heading 30 degrees west of the north direction. To find the horizontal component, we can use the cosine function:
Horizontal component = 360m * cos(60°) ≈ -180m towards the west
To find the vertical component, we can use the sine function:
Vertical component = 360m * sin(60°) ≈ 311.06m towards the north
To find the total displacement, we sum the horizontal and vertical components:
Horizontal displacement = 168.69m - 180m ≈ -11.31m
Vertical displacement = 141.46m + 311.06m ≈ 452.52m
Therefore, the displacement of the boat is ≈ -11.31m [W11.31N] + 452.52m [N].
To calculate the average velocity of the boat, we divide the total displacement by the time taken:
Average velocity = Displacement / Time
Average velocity = (-11.31m[W11.31N] + 452.52m[N]) / 22s
The direction of the average velocity is the same as the direction of the total displacement, which is [N].
To determine the displacement and average velocity of the boat, we need to break down the given distances into their components and calculate the resultant vector.
1. Breaking down the distances:
- Distance in the first leg: 220m [E40N]
- Horizontal component (East): 220m * cos(40°)
- Vertical component (North): 220m * sin(40°)
- Distance in the second leg: 360m [N30W]
- Horizontal component (West): 360m * cos(30°)
- Vertical component (North): 360m * sin(30°)
2. Calculating the individual components:
- Horizontal component (East): 220m * cos(40°) = 220m * 0.766 = 168.5m (rounding to one decimal place)
- Vertical component (North): 220m * sin(40°) = 220m * 0.643 = 141.5m (rounding to one decimal place)
- Horizontal component (West): 360m * cos(30°) = 360m * 0.866 = 311.0m (rounding to one decimal place)
- Vertical component (North): 360m * sin(30°) = 360m * 0.500 = 180.0m
3. Calculating the displacement:
- Horizontal displacement = East - West = 168.5m - 311.0m = -142.5m (since West is considered negative)
- Vertical displacement = North - North = 141.5m + 180.0m = 321.5m
Therefore, the displacement is 142.5m [W21.94N] (since the horizontal component is negative).
4. Calculating the average velocity:
Average velocity = Displacement / Time = 142.5m [W21.94N] / 22s = 6.48m/s [W21.94N] (rounding to two decimal places).
Thus, the displacement of the boat is 142.5m [W21.94N], and the average velocity is 6.48m/s [W21.94N].
To determine the displacement and average velocity of the boat, we need to break down the boat's movement into its horizontal and vertical components.
1. Calculate the horizontal and vertical components of the first leg of the race:
- For the distance traveled in the East direction (horizontal), the horizontal component is 220m * cos(40°).
- For the distance traveled in the North direction (vertical), the vertical component is 220m * sin(40°).
2. Calculate the horizontal and vertical components of the second leg of the race:
- For the distance traveled in the North direction (vertical), the vertical component is 360m * sin(30°).
- For the distance traveled in the West direction (horizontal), the horizontal component is 360m * cos(30°).
3. Add up the horizontal and vertical components separately to find the total horizontal and vertical displacements.
4. The displacement is the resultant vector obtained by combining the horizontal and vertical displacements. Use the Pythagorean theorem to find the magnitude of the displacement, and use trigonometry to determine its direction.
5. Calculate the average velocity using the formula: Average velocity = Displacement / Time.
Let's calculate the displacement and average velocity step by step:
First, let's calculate the horizontal and vertical components of the boat's movement:
Horizontal component of the first leg: 220m * cos(40°) ≈ 167.691m
Vertical component of the first leg: 220m * sin(40°) ≈ 141.189m
Horizontal component of the second leg: 360m * cos(30°) ≈ 311.781m
Vertical component of the second leg: 360m * sin(30°) ≈ 180m
Next, let's calculate the total horizontal and vertical displacements:
Horizontal displacement = (167.691m - 311.781m) ≈ -144.09m
Vertical displacement = (141.189m + 180m) ≈ 321.189m
Using the Pythagorean theorem, we find the magnitude of the displacement:
Displacement magnitude = √((-144.09m)^2 + (321.189m)^2) ≈ 354.563m
To determine the direction, we can use trigonometry:
Displacement direction = tan^(-1)(vertical displacement / horizontal displacement)
Displacement direction ≈ tan^(-1)(321.189m / -144.09m) ≈ -64.157° (measured counterclockwise from the positive x-axis)
Now that we have the displacement (magnitude and direction), we can calculate the average velocity:
Average velocity = Displacement / Time
Average velocity = 354.563m / 22s ≈ 16.116 m/s
Therefore, the displacement of the boat is approximately 354.563m at an angle of -64.157 degrees (measured counterclockwise from the positive x-axis), and the average velocity is approximately 16.116 m/s.