An electronic tracking device is placed on a police dog to monitor its whereabouts relative to the police station. At time t1 = 23 minutes, the dogs placement from the station is 1.2km, 33 degrees north of east. At time t2=57 minutes, the dog's displacement from the station is 2.0km, 75 degrees north of east. Find the magnitude and direction of the dog's average velocity between these two times.
To find the magnitude and direction of the dog's average velocity between time t1 and t2, we need to calculate the dog's displacement and the total time elapsed.
1. Calculate the displacement:
The displacement is given by the difference between the final and initial positions. In this case, the initial position is 1.2km, 33 degrees north of east, and the final position is 2.0km, 75 degrees north of east.
To calculate the displacement in the x (east-west) direction:
- Initial x-displacement = 1.2km * cos(33 degrees)
- Final x-displacement = 2.0km * cos(75 degrees)
- Displacement in x-direction = Final x-displacement - Initial x-displacement
To calculate the displacement in the y (north-south) direction:
- Initial y-displacement = 1.2km * sin(33 degrees)
- Final y-displacement = 2.0km * sin(75 degrees)
- Displacement in y-direction = Final y-displacement - Initial y-displacement
2. Calculate the time elapsed:
The time difference between t1 and t2 is 57 minutes - 23 minutes = 34 minutes.
3. Calculate the average velocity:
The average velocity is the displacement divided by the time elapsed. We can use the formula:
Average velocity = (displacement in x-direction + displacement in y-direction) / elapsed time
4. Calculate the magnitude and direction:
The magnitude of the average velocity can be calculated using the Pythagorean theorem:
Magnitude = sqrt((displacement in x-direction)^2 + (displacement in y-direction)^2)
The direction of the average velocity can be calculated using trigonometry:
Direction = atan((displacement in y-direction) / (displacement in x-direction)), where atan is the arctangent function.
By following this process and using the given information, we can find the magnitude and direction of the dog's average velocity between t1 and t2.
To find the magnitude and direction of the dog's average velocity, we need to calculate the change in displacement and the change in time.
Step 1: Calculate the change in displacement.
The change in displacement is given by the final displacement minus the initial displacement:
Δd = d2 - d1
Given:
d1 = 1.2 km at 33 degrees north of east
d2 = 2.0 km at 75 degrees north of east
We can break down the displacements into their x and y components. The x-component represents the east direction, and the y-component represents the north direction.
For d1:
d1x = d1 * cos(33°)
d1y = d1 * sin(33°)
For d2:
d2x = d2 * cos(75°)
d2y = d2 * sin(75°)
Step 2: Calculate the change in time.
Δt = t2 - t1
Δt = 57 min - 23 min
Δt = 34 min
Step 3: Calculate the average velocity.
Average velocity is given by the change in displacement divided by the change in time:
Vavg = Δd / Δt
Substituting the values we found earlier:
Vavg = (d2x - d1x, d2y - d1y) / Δt
Step 4: Calculate the magnitude and direction.
The magnitude of the average velocity can be found using the Pythagorean theorem:
|Vavg| = √(Vavg_x^2 + Vavg_y^2)
The direction of the average velocity can be found using the inverse tangent function:
θ = tan^(-1)(Vavg_y / Vavg_x)
Now we can calculate the values.
d1x = 1.2 km * cos(33°) ≈ 0.998 km
d1y = 1.2 km * sin(33°) ≈ 0.66 km
d2x = 2.0 km * cos(75°) ≈ 0.532 km
d2y = 2.0 km * sin(75°) ≈ 1.898 km
Δt = 34 min
Vavg_x = (d2x - d1x) / Δt
= (0.532 km - 0.998 km) / 34 min ≈ -0.465 km/min
Vavg_y = (d2y - d1y) / Δt
= (1.898 km - 0.66 km) / 34 min ≈ 0.051 km/min
|Vavg| = √(Vavg_x^2 + Vavg_y^2)
= √((-0.465 km/min)^2 + (0.051 km/min)^2)
≈ 0.466 km/min
θ = tan^(-1)(Vavg_y / Vavg_x)
= tan^(-1)(0.051 km/min / -0.465 km/min)
≈ -6.23°
Therefore, the magnitude of the dog's average velocity between t1 and t2 is approximately 0.466 km/min, and the direction is approximately 6.23 degrees south of west.
To get the coordinates of each point you can use:
(x1, y1) ---> x1=r1*cosA, y1=r1*sinA
(r1 = 1.2km, A = 33deg)
(x2, y2) ---> x2=r2*cosB, y2=r2*sinB
(r2=2.0km, B = 75deg)
The displacement is:
d = sqrt[(x2-x1)^2 + (y2-y1)^2]
The direction of the resultant, angle C is:
C = tan^-1[(y2-y1)/(x2-x1)]
(C is measured from the positive x axis)
***Other trigonometric methods are possible or a scale diagram.