General Question About 2D-Kinematic Problems

[Lori]

I've been doing many Kinematic problems for physics class on like projectiles, falling objects, cars/people/boats,etc.

The thing I'm most confused about is that there are many physics problems involving the x and y directions for velocity, acceleration, position... but like i get confused on whether i should include vector units or not. Is it just good practice to always use vector units (i,j,k)?

For example, when the problem asks for velocity or acceleration of something, how do i know if i should include the vector units?

Another confusing thing when i do these problems is wheter i should use the pythagorean formula and square the x and y components to get the velocity/acceleration vector OR if i am suppose to use the Kinematic equations and plug in numbers and solve.

 

[kuruman]

It depends on what the problem is asking you to find. If it asks you to find a quantity that is a vector like velocity or acceleration, then you cannot go wrong if you write that vector using unit vector notation. So it is good practice. Furthermore, the use of vector notation helps you keep the like components separate from the unlike components and comes in handy when you add vectors. If the problem is asking you to find the magnitude of a vector like speed, that's a clear sign that you must use the Pythagorean theorem.

You can write a single kinematic equation for two dimensional motion, but you have to remember that it is "two equations in one". For example, the position of a projectile at any time t can be written as
$\vec{r}=\vec{v}_0t+\frac{1}{2}\vec{a}t^2$. This is the single equation that is "two equations in one."

Now, $\vec{r} = x\hat{i}+y\hat{j}$, $\vec{v}_0 = v_{0x}\hat{i}+\vec{v}_{0y}\hat{j}$ and $\vec{a}=-g\hat{j}$ so that if we substitute, we get

$x\hat{i}+y\hat{j}=v_{0x}t\hat{i}+v_{0y}t\hat{j}-\frac{1}{2} gt^2 \hat{j}$

The last is a vector equation that says that the vector on the left is the same as the vector on the right. Two vectors are the same when their components are the same. This means that whatever multiplies i-hat on the left must equal whatever multiplies i-hat on the right. Likewise for j-hat. This means we have two equations, namely
$x=v_{0x}t$
$y=v_{0y}t-\frac{1}{2} gt^2$
which are the familiar kinematic equations for a projectile shot from the origin. I hope this helps.

 

[Lori]

Thank you so much. Also, I think the last two equations you mentioned are what confused me too. The first equation you showed it worked for two dimensions. X = X0 +VoxT +1/2 ax t^2 also works to find Y components for velocity and acceleration right? So you are saying those last two equations are just simpler equations instead of this except one only works for one axis?

Or do those last 2 equations only work for projectile where x is range and y is height of projectile

 

[kuruman]

Or do those last 2 equations only work for projectile where x is range and y is height of projectile

Yes, these are the general 2-D equations adapted to projectile motion with the particle projected from the origin. I used the simpler example of projectile motion to illustrate the method. The most general equation for 2-D motion under constant acceleration is
$\vec{r}=\vec{r}_0+\vec{v}_0t+\frac{1}{2}\vec{a}t^2$
where $\vec{r}=x\hat{i}+y\hat{j}$, $\vec{r}_0=x_0\hat{i}+y_0\hat{j}$, $\vec{v}_0=v_{0x}\hat{i}+v_{0y}\hat{j}$ and $\vec{a}=a_x\hat{i}+a_y\hat{j}$
which implies the two equations
$x=x_0+v_{0x}t+\frac{1}{2} a_x~t^2$
$y=y_0+v_{0y}t+\frac{1}{2} a_y~t^2$

 

[Dr.D]

Whether to use vector notation depends first and foremost on what is asked in the problem. If you are working independently (not a problem assigned by someone else), then use what makes sense. In this last regard, vector notation is often not the best choice, particularly if you will need to write computer code to implement your calculations. For the last case, matrix notation is often far superior as it is easily programmed in computer code.

I started college shortly after the Russians launched the first Earth satellite, Sputnik. There was a general panic in American schools, saying, "we're behind; we have to catch up with the Russians." One of the major steps in "catching up" was to go to vector notation for everything, so that is how I first learned mechanics. As I advanced and began to work more complex problems, I learned about energy methods, particular the Lagrange formulation, and these always lead to scalar equations. Since this is the approach I employ for almost everything today, I rarely use vector notation (I did so about 10 days ago, for the first time in ages, despite working in mechanics daily).

So the lesson I would suggest to you is simply, do what makes sense. If you want vector equations, use vector notation. If you want scalar equations (for computer work), use scalars. Think ahead to what you want to have in the end, and go that route.