Question

A hiker travels at 4 miles per hour at a heading of S 35° E from a ranger station. After 3
hours how far south and how far east is the hiker from the station?

Answer 1

Answer:

First, let's define our coordinates.

Let's call East as the positive x-axis and North as the positive y-axis.

Now the data that we have is:

Speed  = 4 mph

Time = 3 hours.

Then the total distance traveled is:

D = 4mph*3h = 12 miles.

We also know that the angle is S 35° E.

(from east, 35° to south)

So we can write this angle as -35°.

Then if we want to know the component in South and East, we can first think this in polar coordinates

(R, θ)

where R = 12 miles, and θ = -35°

Now with this, we have that:

X = R*cos(θ) = the displacement to the east.

Y = R*sin(θ) = the displacement to the north (then -Y is the displacement to the south).

Then we have:

X = 12mi*cos(-35°) = 9.83mi

Y = 12mi*sin(-35°) = - 6.89 mi.

Then:

The displacement to the East is X = 9.83 miles

The displacement to the South is -Y = 6.89 miles.

Answer 2

The magnitude of southern displacement of the hiker is 6.88 miles.

The magnitude of eastern displacement of the hiker is 9.83 miles.

The given parameters;

• Velocity of the hiker, v = 4 mph
• Position of the hiker, θ = S 35⁰ E
• Time of motion, t = 3 hours

The resultant displacement of the hiker is calculated as follows;

D = vt

D = 4 mph x 3 h

D = 12 miles

The magnitude of southern displacement of the hiker is calculated as follows;

$D_y = D \times sin(\theta)\\\\D_y = 12 \ miles \times sin(35)\\\\D_y = 6.88 \ miles$

The magnitude of eastern displacement of the hiker is calculated as follows;

$D_x = D \times cos(35)\\\\D_x = 12 \ miles \times cos(35)\\\\D_x = 9.83 \ miles$

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