VQ - Problem Solving with Vectors Lesson

Problem Solving with Vectors

So now that we have learned about the different parts of a vector, let's apply those skills to solve some real-world problems.

Example 1

James is pushing the handle of a lawn mower with a force of 450 newtons at an angle of 56° with the ground. Find the magnitude of the horizontal and vertical components of the force.

man pushing lawnmower with x: horizontal component and y: vertical component indicated text annotation indicator

We'll use cosine and sine to find the magnitude of the horizontal and vertical components:

Horizontal	Vertical
$LaTeX: \cos56°=\frac{\left\|x\right\|}{450}$ $\cos56°=\frac{\left\|x\right\|}{450}$	$LaTeX: \sin56°=\frac{\left\|y\right\|}{450}$ $\sin56°=\frac{\left\|y\right\|}{450}$
$LaTeX: \left\|x\right\|=450\cos56°$ $\left\|x\right\|=450\cos56°$	$LaTeX: \left\|y\right\|=450\sin56°$ $\left\|y\right\|=450\sin56°$
$LaTeX: \left\|x\right\|=251.6\:Newtons$ $\left\|x\right\|=251.6\:Newtons$	$LaTeX: \left\|y\right\|=373.1\:Newtons$ $\left\|y\right\|=373.1\:Newtons$

For each problem below, find the magnitude of the horizontal and vertical components. Hint: start by sketching a picture!

1. A player kicks a soccer ball so that it leaves the ground with a velocity of 44 feet per second at an angle of 33° with the ground.

Solution: $LaTeX: \left|x\right|=36.9\frac{ft}{s};\:\left|y\right|=240\frac{ft}{s}$ $\left|x\right|=36.9\frac{ft}{s};\:\left|y\right|=240\frac{ft}{s}$

2. While digging in his garden, James pushes a shovel into the ground with a force of 630 Newtons at and angle of 70° with the ground.

Solution: $LaTeX: \left|x\right|=215.5N;\:\left|y\right|=592N$ $\left|x\right|=215.5N;\:\left|y\right|=592N$

Example 2

Watch this video to try another problem.

Let's try a few problems:

1. An airplane is traveling due east with a speed of 600 mi/hr. The wind blows at 85 mi/hr at a direction of 121°. Determine the ground speed and true course of the airplane.

Solution: 674.28mph, 95°

2. Marge is pulling a sled with a force of 275 N by holding a rope at an angle of 58°. Her brother is pushing the sled from behind to give a horizontal force of 320 N. Determine the magnitude and direction of the resultant force on the sled.

Solution: 520.9N, 26.6°

3. Lane is throwing the javelin in a track meet. While running at 4 m/s she throws the javelin with a velocity of 28 m/s at an angle of 48°. Determine the resultant speed and direction of the javelin?

Solution: 30.8m/s, 42.5°

IMAGE SOURCE: GOOGLE IMAGES

Horizontal	Vertical
$LaTeX: \cos56°=\frac{\left\|x\right\|}{450}$ $\cos56°=\frac{\left\|x\right\|}{450}$	$LaTeX: \sin56°=\frac{\left\|y\right\|}{450}$ $\sin56°=\frac{\left\|y\right\|}{450}$
$LaTeX: \left\|x\right\|=450\cos56°$ $\left\|x\right\|=450\cos56°$	$LaTeX: \left\|y\right\|=450\sin56°$ $\left\|y\right\|=450\sin56°$
$LaTeX: \left\|x\right\|=251.6\:Newtons$ $\left\|x\right\|=251.6\:Newtons$	$LaTeX: \left\|y\right\|=373.1\:Newtons$ $\left\|y\right\|=373.1\:Newtons$