Physics Practice Problems Vector Addition & Subtraction Practice Problems Solution: You leave the airport in College Station and fly 2...

Solution: You leave the airport in College Station and fly 24.0 km in a direction 34.0° south of east. You then fly 46.0 km  m kmdue north.How far must you then fly to reach a private landing strip that is 32.0 km due west of the College Station airport?In what direction?

Problem

You leave the airport in College Station and fly 24.0 km in a direction 34.0° south of east. You then fly 46.0 km due north.

How far must you then fly to reach a private landing strip that is 32.0 km due west of the College Station airport?

In what direction?

Solution

Steps for finding a missing vector:

  1. Write the vector equation for the problem (drawing a diagram usually helps a lot!)
  2. Decompose the given vectors and organize known information in a table.
  3. Write algebraic equations and solve.
  4. Find magnitude and angle of the resultant if required.

When writing a vector equation for one of these problems, it'll be in a form like 

A+B+...=R,

where the missing vector is one of the vectors on the left-hand side of the equation. Note that if the problem is about returning to the starting point, R=0.

For decomposing given vectors, we'll have to define a coordinate system if the problem hasn't, then use the equations

Ax=|A| cos θAy=|A| sin θ

If the final answer is supposed to be in magnitude-angle notation, we'll need to use these equations to get the magnitude and direction:

|A|=Ax2+Ay2

tan θ=AyAx

Step 1. This problem has the plane flying three separate legs (24.0 km at 34.0° south of east, 46.0 km north, and the unknown). We'll call them A, B, and C in order. The resultant R is 32.0 km west. So our vector equation is

A+B+C=R

Step 2. Define a coordinate system so that north is the +y-direction and east is the +x-direction. Then we can decompose vector A into x- and y-components. "South of east" in this coordinate system is counterclockwise from the +x-axis, so

Ax=|A| cos θ=(24.0 km) cos (-34.0°)=19.90 km  Ay=|A| sin θ=(24.0 km) sin (-34.0°)=-13.42 km

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