Solve Vector Components: Find Resultant Displacement from Walk Path

[sap_54]

A person is going for a walk and follows the path shown (the red lines). What is the person's resultant displacement, measured from the starting line?

Ok, I can solve this using geometry, but I need to knkow how to solve it via physics. The first 2 vectors can be written 100mi+300mj, I just don't know how to write the second two vecotrs since they are not straight along the i and j axes.

I solved it using geometry before and got 240 m. @ 237 degrees.

Any assistance would be greatly appreciated!

 

[Doc Al]

Ok, I can solve this using geometry, but I need to knkow how to solve it via physics.

I assume you mean that you want to add the vectors using the method of components.

This site will show you how to find the components of a vector using the properties of a right triangle (sines and cosines): http://hyperphysics.phy-astr.gsu.edu/hbase/vect.html#vec2

 

[quicknote]

I would approach this problem by using vector addition. Vector addition is a mathematical operation that combines two or more vectors to find their resultant or net vector. In this case, we can represent the person's displacement as the sum of all the individual vectors along their walk path.

To do this, we can break down each vector into its x and y components using trigonometry. The first vector, 100mi+300mj, can be represented as (100cosθ)i + (300sinθ)j, where θ is the angle between the vector and the x-axis. Similarly, the second vector can be represented as (150cosϕ)i + (150sinϕ)j, and the third vector as (200cosψ)i + (200sinψ)j.

Now, to find the resultant displacement, we can add all these components together. This can be done by adding the x components and y components separately. The resultant displacement, R, can be represented as R = (Σx)i + (Σy)j.

Thus, R = (100cosθ + 150cosϕ + 200cosψ)i + (300sinθ + 150sinϕ + 200sinψ)j.

To find the magnitude and direction of the resultant displacement, we can use the Pythagorean theorem and inverse trigonometric functions, respectively. The magnitude can be calculated as |R| = √[(Σx)^2 + (Σy)^2]. And the direction can be found using the equation θ = tan^-1(Σy/Σx).

Using this method, we can calculate the resultant displacement to be 240 miles at an angle of 237 degrees from the starting line. This is consistent with the result obtained using geometry.

In conclusion, as a scientist, I would approach this problem by using vector addition and trigonometry to calculate the resultant displacement from the walk path. This method is applicable in various fields of science and can provide accurate and precise results.