three children are throwing a frisbee in a park. the first child throws the frisbee 32m[W14S] to the second child who then throws it 15m[E62S] to the third child. what is the total displacement of the frisbee?
To find the total displacement of the frisbee, we need to find the resultant vector of the two individual displacements.
First, let's break down the given displacements into their horizontal (east-west) and vertical (north-south) components.
For the first child's throw:
- Horizontal component: 32m * cos(14°) = 31.2199m (rounded to 4 decimal places) to the west.
- Vertical component: 32m * sin(14°) = 8.0923m (rounded to 4 decimal places) to the south.
For the second child's throw:
- Horizontal component: 15m * cos(62°) = 6.4264m (rounded to 4 decimal places) to the east.
- Vertical component: 15m * sin(62°) = 12.9590m (rounded to 4 decimal places) to the south.
To find the total horizontal displacement, we add the horizontal components together:
31.2199m (first child's throw to the west) + 6.4264m (second child's throw to the east) = 37.6463m (rounded to 4 decimal places) to the west.
Similarly, to find the total vertical displacement, we add the vertical components together:
8.0923m (first child's throw to the south) + 12.9590m (second child's throw to the south) = 21.0513m (rounded to 4 decimal places) to the south.
Finally, we can find the magnitude and direction of the total displacement using Pythagorean theorem and trigonometry:
Magnitude: sqrt((37.6463m)^2 + (21.0513m)^2) = 42.8319m (rounded to 4 decimal places).
Direction: arctan(21.0513m / 37.6463m) ≈ 28.7296° south of west (rounded to 4 decimal places).
Therefore, the total displacement of the frisbee is approximately 42.8319m at an angle of 28.7296° south of west.
To find the total displacement of the frisbee, we need to add the individual displacements of each throw.
1. The first child throws the frisbee 32m[W14S].
- The "W" indicates a west direction and "14S" indicates a 14-degree angle south of the west.
- This means the frisbee moves 32m to the west and at an angle of 14 degrees south of the west.
2. The second child throws the frisbee 15m[E62S].
- The "E" indicates an east direction and "62S" indicates a 62-degree angle south of the east.
- This means the frisbee moves 15m to the east and at an angle of 62 degrees south of the east.
To calculate the total displacement, we can break down each displacement into horizontal (east/west) and vertical (north/south) components and then add them up:
Horizontal Displacement:
- The first child throws the frisbee 32m to the west, so the horizontal displacement is -32m.
- The second child throws the frisbee 15m to the east, so the horizontal displacement is +15m.
Vertical Displacement:
- In the first throw, the frisbee moves 14 degrees south of the west, which means it has a vertical displacement of -32m*sin(14) ≈ -8.27m.
- In the second throw, the frisbee moves 62 degrees south of the east, which means it has a vertical displacement of -15m*sin(62) ≈ -13.18m.
Now we can calculate the total displacement:
Horizontal Displacement = -32m + 15m = -17m (to the west)
Vertical Displacement = -8.27m + (-13.18m) ≈ -21.45m (to the south)
The total displacement of the frisbee is approximately 17m to the west and 21.45m to the south.
To find the total displacement of the frisbee, we need to combine the individual displacements of each throw. The displacements are given in terms of magnitude (distance) and direction.
Let's break down the given displacements:
1. The first child throws the frisbee 32m to the west at an angle of 14 degrees south of west.
2. The second child throws the frisbee 15m to the east at an angle of 62 degrees south of east.
To calculate the total displacement, we will add the individual displacements using vector addition. To do this, we convert the polar coordinates to Cartesian coordinates.
For the first child's throw:
The x-component (west-east direction) of the first throw can be calculated using trigonometry:
x₁ = 32m * cos(14°) [negative because it's west]
x₁ ≈ -31.307m
The y-component (south-north direction) of the first throw can also be calculated using trigonometry:
y₁ = 32m * sin(14°) [negative because it's south]
y₁ ≈ -8.507m
So, the first child's throw can be represented as a vector: (-31.307m, -8.507m)
For the second child's throw:
The x-component (west-east direction) of the second throw can be calculated using trigonometry:
x₂ = 15m * cos(62°) [positive because it's east]
x₂ ≈ 6.037m
The y-component (south-north direction) of the second throw can also be calculated using trigonometry:
y₂ = 15m * sin(62°) [negative because it's south]
y₂ ≈ -13.073m
So, the second child's throw can be represented as a vector: (6.037m, -13.073m)
To find the total displacement, we need to add the x-components and y-components separately:
Total x-component = x₁ + x₂
Total x-component = -31.307m + 6.037m
Total x-component ≈ -25.27m
Total y-component = y₁ + y₂
Total y-component = -8.507m + (-13.073m) [negative because the second throw goes south]
Total y-component ≈ -21.58m
Therefore, the total displacement of the frisbee is approximately (-25.27m, -21.58m) or about 33.74m at an angle of approximately 136 degrees south of west.