A person walks 25m west and then 45m at an angle of 60 degrees north of east. What is the magnitude of the total displacement?
North component of resultant: 45 sin 60 = 38.97 m
East component of resultant: 45 cos 60 - 25 = -2.5 m
displacement magnitude
= sqrt[(38.97)^2 + (-2.5)^2]
= 39.05 m
39.05
Well, if you ask me, that person seems a bit lost on their way to the circus! But let me check my clown compass for you.
To calculate the total displacement, we need to find the resultant vector by adding the individual vectors. So, let's break it down.
The first vector is 25 meters west, which we can represent as -25i (with 'i' being the westward direction).
The second vector is 45 meters at an angle of 60 degrees north of east. To break it down, we can use a little trigonometry. The vertical component is 45 * sin(60°), which is approximately 38.91m. The horizontal component is 45 * cos(60°), which is approximately 22.50m. Representing these two components as 38.91j (northward) and 22.50i (eastward), respectively.
Now, time to add these vectors together:
Resultant vector = -25i + 22.50i + 38.91j
After summing them up, we get approximately 44.50i + 38.91j.
To find the magnitude of the total displacement, we use the Pythagorean theorem:
Magnitude = sqrt((44.50)^2 + (38.91)^2) meters
So, all calculations considered, the magnitude of the total displacement is approximately 58.87 meters.
To find the total displacement, we need to calculate the horizontal and vertical components separately and then find the magnitude.
Let's start by breaking down the displacements into their horizontal and vertical components.
Horizontal component:
The person walks 25m west, which means their horizontal displacement is -25m. (We take west as the negative x-direction.)
Vertical component:
The person walks 45m at an angle of 60 degrees north of east. To find the vertical component, we need to calculate the sine of the angle and multiply it by the magnitude.
Vertical displacement = Magnitude * sin(angle)
Vertical displacement = 45m * sin(60°)
Vertical displacement ≈ 45m * 0.866 ≈ 38.97m
Now, let's sum up the horizontal and vertical components:
Horizontal displacement = -25m
Vertical displacement ≈ 38.97m
To find the total displacement, we'll use the Pythagorean theorem:
Total displacement = sqrt((Horizontal displacement)^2 + (Vertical displacement)^2)
Total displacement = sqrt((-25m)^2 + (38.97m)^2)
Total displacement ≈ sqrt(625m^2 + 1520.4m^2) ≈ sqrt(2145.4m^2) ≈ 46.34m
Therefore, the magnitude of the total displacement is approximately 46.34m.
To find the magnitude of the total displacement, we can treat the two displacements as vectors and then find the resultant vector.
Let's break down the given information into vectors:
- The person walks 25m west. This can be represented by a vector pointing towards the west with a magnitude of 25m.
- The person then walks 45m at an angle of 60 degrees north of east. This can be represented by a vector pointing towards the northeast with a magnitude of 45m.
Since we have the magnitudes and directions of the two vectors, we can now find the resultant vector by adding the two vectors together.
To add vectors, we need to resolve them into their horizontal and vertical components.
For the vector pointing west with a magnitude of 25m, the horizontal component is -25m (negative because it's pointing towards the left) and the vertical component is 0m (no vertical movement).
For the vector pointing northeast with a magnitude of 45m, we can consider it as the sum of a vector pointing east and a vector pointing north. The angle of 60 degrees north of east can be split into a 30 degrees angle for each component.
Using trigonometry, we can find the horizontal and vertical components of this vector.
- The horizontal component is 45m * cos(30 degrees) = 45m * 0.866 = approximately 38.97m.
- The vertical component is 45m * sin(30 degrees) = 45m * 0.5 = 22.5m.
Now, we can add the horizontal components and vertical components separately to find the resultant vector.
Horizontal component: -25m + 38.97m = 13.97m
Vertical component: 0m + 22.5m = 22.5m
Using the Pythagorean theorem, we can find the magnitude of the resultant vector:
Magnitude = sqrt((horizontal component)^2 + (vertical component)^2)
Magnitude = sqrt((13.97m)^2 + (22.5m)^2)
Magnitude ≈ sqrt(195.16 + 506.25) ≈ sqrt(701.41) ≈ 26.49m
So, the magnitude of the total displacement is approximately 26.49m.