Physics question

Question

A diver explores a shallow reef off the coast of Belize. She initially swims d⃗ 1

= 85.5 m north, makes a turn to the east and continues for d⃗ 2

= 159.2 m, then follows a big grouper (a kind of fish) for d⃗ 3

= 82.6 m in the direction θ

= 38.4° north of east. Finally, she swims d⃗ 4

= 192.1 m south.

What is the straight line distance in meters between her initial and final positions? In what direction should she swim? Express your answer as an angle in degrees north of west.

Answer 1

With these sorts of questions, I like to think of it from a component perspective. You have components that contribute to your North to South direction (or y-axis) and components that contribute to your East to West direction (or x-axis). Looking at the distances that are directly North and South (assuming North is positive and South is negative) we have:

d1 = 85.5 meters

d4 = -192.1 meters (negative because we are assuming south is negative)

While these are the distances that are explicitly stated, there is still another distance (d3) that has a components that contribute to the total distance both in the North to South axis and the East to West axis. This is due to the travel direction being at an angle 38.4° north of east. Starting with the y component, we can calculate it by doing d3*sin(38.4). We will call this d3_y to represent the y component of d3. Performing this calculation gives us d3_y = 51.3 meters.

Our total distance in the North to South direction (or y-axis) is given by:

D_y = d1 + d4 + d3_y (just the y component!) = (85.5) + (-192.1) + (51.3) = -55.3 meters

Remember that we assigned positive as North, so a negative distance denotes that we are heading South in the North to South direction.

We can do a similar thing for the East to West distances. Assuming East is positive and West is negative, we have:

d2 = 159.2 meters (This is the only component that is directly East or West).

Note that d3 also has a component that contributes to the East to West direction as well.

d3_x = d3*cos(38.4) = 64.7 meters.

D_x = d2 + d3_x (just the x component!) = 159.2 + 64.7 = 223.8 meters

From a graphical perspective we have moved 223.8 meters in the positive x (East) direction and 55.3 in the negative y direction (South). This means she is currently South-East of where she started. The straight line distance can be calculated using the Pythagorean theorem √ (-55.3²) + (223.8)² = 230.5 meters

To calculate the angle, we can take the inverse tangent arctan(55.3/223.8) = 13.9 degrees South of East. (NOT THE FINAL ANSWER)

The questions states that it wants the answer in terms of degrees North of West. Think of going directly West as being 0 degrees. Going clockwise until we reach our direction of 13.9 degrees South of East, we get an angle of 193.9 degrees North of West

Final answer: 230.5 meters at 193.9 degrees North of West

As far as what direction she should swim (assuming it means to get back to her initial position), she should swim North West at 13.9 degrees