You decide to take your dog to the leash-free zone
in the park. While playing, the dog runs after a ball
and heads 24 m [W 12° S] and then gets distracted
by a squirrel and runs 33 m [E 52° S]. Determine the
displacement of the dog
31 m (W 84° S)
To determine the displacement of the dog, we need to find the vector sum of the two displacements:
1. Start by breaking down the first displacement into its north-south and east-west components.
The direction W 12° S can be broken down into:
- North-South Component: 24 m * sin(12°S)
- East-West Component: 24 m * cos(12°S)
Substituting the values:
North-South Component = 24 m * sin(12°) ≈ -4.91 m
East-West Component = 24 m * cos(12°) ≈ 23.52 m
So, the first displacement can be represented as:
24 m [23.52 m E, -4.91 m]
2. Similarly, break down the second displacement into its north-south and east-west components.
The direction E 52° S can be broken down into:
- North-South Component: 33 m * sin(52°S)
- East-West Component: 33 m * cos(52°S)
Substituting the values:
North-South Component = 33 m * sin(52°) ≈ -25.25 m
East-West Component = 33 m * cos(52°) ≈ 19.36 m
So, the second displacement can be represented as:
33 m [19.36 m E, -25.25 m]
3. Now, add the two displacements together to find the resultant displacement.
Adding the east-west components:
23.52 m E + 19.36 m E = 42.88 m E
Adding the north-south components:
-4.91 m + (-25.25 m) ≈ -30.16 m
So, the resultant displacement is approximately:
42.88 m E, -30.16 m
Therefore, the displacement of the dog is approximately 42.88 m east and 30.16 m south.
To determine the displacement of the dog, we need to find the overall change in position from the starting point to the ending point.
Step 1: Convert the distances given in both directions (west and east) to their respective x-axis values.
Since the dog heads 24 m [W 12° S], we know that it moves 24 m towards the west. Since it also moves 12° south of the west, we can use trigonometry to determine the x-axis value. The x-axis value, in this case, is given by:
x-west = 24 m * sin(12°)
Step 2: Convert the distances given in both directions (north and south) to their respective y-axis values.
Since the dog gets distracted and runs 33 m [E 52° S], we know that it moves 33 m towards the east. Since it also moves 52° south of the east, we can again use trigonometry to determine the y-axis value. The y-axis value, in this case, is given by:
y-south = 33 m * sin(52°)
Step 3: Determine the overall displacement by using the Pythagorean theorem.
The overall displacement is given by the square root of the sum of the squares of the x-axis and y-axis values. Mathematically, it can be represented as:
displacement = √(x-west² + y-south²)
Now we can plug in the values to calculate the displacement.