Two nomads leave camp at the same time one walks at 5km/h on a bearing 039°other walks at 7.5km/h on a bearing 265° after 2hrs,how far apart are they and what is the bearing of the second from the first
23
To find the distance between the two nomads after 2 hours, we can use the formula:
Distance = Speed x Time
Nomad 1 walks at 5 km/h, so the distance Nomad 1 covers in 2 hours is:
Distance 1 = 5 km/h x 2 h = 10 km
Nomad 2 walks at 7.5 km/h, so the distance Nomad 2 covers in 2 hours is:
Distance 2 = 7.5 km/h x 2 h = 15 km
To find the bearing of the second nomad from the first, we can subtract the bearing of the first nomad from 180° and add the result to the bearing of the second nomad.
Bearing of first nomad: 039°
Bearing of second nomad: 265°
Bearing of second nomad from the first = (180° - 39°) + 265° = 406°
Therefore, after 2 hours, the two nomads are approximately 10 km apart, and the bearing of the second nomad from the first is 406°.
To find the distance between the two nomads and the bearing of the second nomad from the first, we can use the concept of vector addition.
First, let's break down the motion of each nomad into its northward (y-component) and eastward (x-component) components.
For the first nomad walking at a speed of 5 km/h on a bearing of 039°, we can calculate the components using trigonometry. The northward component is given by:
5 km/h * sin(39°) ≈ 5 km/h * 0.6293 ≈ 3.1465 km/h
And the eastward component is given by:
5 km/h * cos(39°) ≈ 5 km/h * 0.7692 ≈ 3.8462 km/h
Similarly, for the second nomad walking at a speed of 7.5 km/h on a bearing of 265°:
The northward component:
7.5 km/h * sin(265°) ≈ 7.5 km/h * (-0.2588) ≈ -1.941 km/h
The eastward component:
7.5 km/h * cos(265°) ≈ 7.5 km/h * (-0.9659) ≈ -7.2442 km/h
Next, we need to find the total displacement for each nomad after 2 hours of motion. To do this, we multiply each component by the time (2 hours in this case).
For the first nomad:
Northward displacement = 3.1465 km/h * 2 h = 6.293 km
Eastward displacement = 3.8462 km/h * 2 h = 7.6924 km
For the second nomad:
Northward displacement = -1.941 km/h * 2 h = -3.882 km
Eastward displacement = -7.2442 km/h * 2 h = -14.4884 km
Now, we can find the total displacement vector by adding the displacements of the two nomads.
The northward component of the total displacement = 6.293 km - 3.882 km = 2.411 km
The eastward component of the total displacement = 7.6924 km - 14.4884 km = -6.796 km
To find the distance between the two nomads, we can use the Pythagorean theorem:
Distance = √((northward component)^2 + (eastward component)^2)
Distance = √((2.411 km)^2 + (-6.796 km)^2) ≈ √(5.810681 km^2 + 46.188736 km^2)
Distance ≈ √(52.999417 km^2) ≈ 7.279 km
The distance between the two nomads is approximately 7.279 km.
To find the bearing of the second nomad from the first, we can use the inverse tangent function (arctan).
Bearing = arctan((eastward component of the total displacement) / (northward component of the total displacement))
Bearing = arctan((-6.796 km) / (2.411 km)) ≈ -70.709°
Therefore, the bearing of the second nomad from the first is approximately 70.709°.