Using Vectors to Determine Projectile Position

In summary, to determine the position of a projectile using vectors, you can use the position vector formula \vec{r} = x(t)\hat{i} + y(t)\hat{j}. This can be achieved by first finding the acceleration, then using integrals to calculate the velocity and position at any given time. It is important to project each vector along the x and y directions using the triangle-equalities. This process is essential in solving for both the velocity and position of the projectile in two dimensions. The complexity of the force does not affect this process, as the same procedure can be applied.
  • #1
amcavoy
665
0
In rectangular coordinates you have:

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

[tex]x=v_{0}\cos{(\theta)}t[/tex]

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.
 
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  • #2
The position vector r of the object at time t is just:

[tex]\vec{r} = x(t)\hat{i} + y(t)\hat{j}[/tex]

where the i and j are unit vectors in the x and y directions.
 
  • #3
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

[tex]\vec{F} = F_x \vec{e_x} + F_y \vec{e_y} = m(a_x \vec{e_x} + a_y \vec{e_y})[/tex]

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

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

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 :

[tex]F_x = ma_x [/tex]
[tex]v_x = v_{0x} + a_xt[/tex] and
[tex]x = r_{0x} + v_{0x}t+ a_x \frac{t^2}{2}[/tex]

[tex]v_{0x} = ||\vec{v_0}||cos( \theta)[/tex]
[tex]r_{0x} = ||\vec{r_0}||cos( \theta)[/tex]

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 [tex]\vec{A} = A_x \vec{e_x} + A_y \vec{e_y}[/tex]

[tex]A_x = ||\vec{A}||cos( \theta)[/tex]
[tex]A_y = ||\vec{A}||sin( \theta)[/tex]

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
 

Related to Using Vectors to Determine Projectile Position

What is a vector?

A vector is a mathematical representation of a quantity that has both magnitude (size) and direction. It is usually represented by an arrow, with the length of the arrow representing the magnitude and the direction of the arrow representing the direction.

How are vectors used in determining projectile position?

Vectors are used to represent the velocity and acceleration of a projectile, which are both crucial in determining its position at any given time. By breaking down the velocity and acceleration vectors into their horizontal and vertical components, we can calculate the position of a projectile at any point in its trajectory.

What is the difference between displacement and distance in projectile motion?

Displacement refers to the straight-line distance and direction between an object's starting and ending points, while distance is the total length of the path traveled by the object. In projectile motion, the displacement vector will always be equal to the distance traveled, as the path of the projectile is a straight line.

How does air resistance affect projectile motion?

Air resistance, also known as drag, is a force that acts in the opposite direction of an object's motion through air. This force can cause a decrease in the projectile's velocity and a change in its trajectory, ultimately affecting its position. In most cases, air resistance can be ignored in projectile motion calculations, but it becomes more significant at higher velocities and longer distances.

What are some real-world applications of using vectors to determine projectile position?

Vectors are used in various fields, such as physics, engineering, and sports, to analyze and predict the motion of projectiles. Some examples include predicting the trajectory of a ball in sports like baseball and golf, calculating the trajectory of a rocket or missile, and analyzing the motion of objects launched from a catapult or slingshot.

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