Using Vectors to Determine Projectile Position

[amcavoy]

In rectangular coordinates you have:

$$y=-\frac{1}{2}gt^2+v_{0}\sin{(\theta)}t+h$$

$$x=v_{0}\cos{(\theta)}t$$

Is there any way to use vectors to determine the postion of a projectile? If so, how would you convert rectangular coordinates to vectors?

Thanks.

 

[James R]

The position vector r of the object at time t is just:

$$\vec{r} = x(t)\hat{i} + y(t)\hat{j}$$

where the i and j are unit vectors in the x and y directions.

 

[marlon]

Alex,

What you are asking about is the very base of Newtonian mechanics. It works like this : Suppose we work in two dimensions denoted by a x-axis and an y-axis. You can work in as many dimensions as you want because all you have to do is add a unit vector to the formula's, as you will see.

Starting from the acceleration ,one can calculate the velocity and position by using integrals at any time : r_0 is initial position at t=0, v_0 is initial velocity

$$\vec{F} = F_x \vec{e_x} + F_y \vec{e_y} = m(a_x \vec{e_x} + a_y \vec{e_y})$$

Where the e_x and e_y denoted the x and y-direction (ie the unit vectors)

Now, integrating will yield
$$\vec{v} = \vec{v_0} + \vec{a}t$$
$$\vec{r} = \vec{r_0} + \vec{v_0}t+ \vec{a} \frac{t^2}{2}$$

Now, the trick really is (and that's the essential part) to apply the same procedure in each direction. The procedure i am talking about is projecting each vector along a direction using the triangle-equalities.

For example : in the x-direction you will have :

$$F_x = ma_x$$
$$v_x = v_{0x} + a_xt$$ and
$$x = r_{0x} + v_{0x}t+ a_x \frac{t^2}{2}$$

$$v_{0x} = ||\vec{v_0}||cos( \theta)$$
$$r_{0x} = ||\vec{r_0}||cos( \theta)$$

the ||.|| denotes the MAGNITUDE of the vector

You see ? the clue is that each vector can be written as a sum of an x and y component $$\vec{A} = A_x \vec{e_x} + A_y \vec{e_y}$$

$$A_x = ||\vec{A}||cos( \theta)$$
$$A_y = ||\vec{A}||sin( \theta)$$

You will need to be carefull with the signs of the x and y components because those depend of the direction of each component with respect to the actual x and y axis.

So, when a force is given, like : F = m(2e_x - 9.81e_y) and the initial position has components r_0 = 2e_x + 6e_y and the initial velocity v_0 = 6e_y, can you write down the equations for both velocity and position in each direction ?

In the end , you must realize that this system is very easy because, nomatter how complicated the force may look, the procedure to determin both position and velocity as a function of time is always the same: Projecting the vectors along the given directions. Once, you have done that, you can do almost anything with these formula's...

regards
marlon