Suppose that a particle with mass m moving on a plane having position vector
r(t)=a cos(t/4)i + a sin(t/4)j ,
where t>0 is the time and a>0 is any constant.
Find the equation of the path that the particle is moving.
My question: what are required to find here?
Can we substitute t=0 to the above position vector( say r(0) ), find another position of the particle and subtract r(0) from r(t) and simply get the path?
They ask you to
"Find the equation of the path that the particle is moving."
??? They just gave you the equation of the path, right?
recall that the equation of a circle is
x = r cosθ
y = r sinθ
your function r(t) is just a circle of radius a with center at (0,0)
where θ = t/4
So as for the question(the path which the particle is moving), can I simply write the answer as it is a circle with radius a with center at (0,0) where theta=t/4 ?
good question. They asked for the equation.
I suspect that maybe they did just want a description of the path.
¿Quien sabe?
Or can we substitute t=0 to the position vector(say r0) and find another position of the particle and substrct r0 from r(t) and obtain the path?
(As we do when finding a equation of a line)
To find the equation of the path that the particle is moving, we need to eliminate the parameter t from the position vector equation. This will give us an equation that relates the x and y coordinates of the particle on the plane.
Substituting t = 0 into the position vector r(t) will give you the initial position of the particle, which is r(0). However, subtracting r(0) from r(t) will give you the displacement vector of the particle from its initial position, not the equation of the path.
To eliminate the parameter t and find the equation of the path, we can use the trigonometric identity cos^2(t) + sin^2(t) = 1. In this case, we have cos(t/4)^2 + sin(t/4)^2 = 1.
Simplifying this expression, we get:
(a cos(t/4))^2 + (a sin(t/4))^2 = a^2 [cos^2(t/4) + sin^2(t/4)] = a^2
Now, let's rewrite the position vector r(t) using the identity cos^2(t/4) + sin^2(t/4) = 1:
r(t) = a cos(t/4)i + a sin(t/4)j
= a cos(t/4)(cos(t/4)i + sin(t/4)j) + a sin(t/4)(cos(t/4)i + sin(t/4)j)
= a (cos^2(t/4) + sin^2(t/4))(i + j)
= a (1)(i + j)
= a (i + j)
Therefore, the equation of the path that the particle is moving is:
x = y
This means that the particle is moving along a straight line in the plane, where the x-coordinate is equal to the y-coordinate.