One Directional Motion

There are lots of different types of motion, angular, several dimensional, and some other possibilities, but the most basic one to prep you for the other would have to be one directional motion.

What is One Directional Motion?

When the calculations we make on an object are only based on their movement in one direction, as in we only focus on that rather than any other aspects, we can figure out many things such as it’s position, velocity, and acceleration

We’re going to use a point for our explanation, the point would have infinitesimal mass, so it can just be ignored. Do remember that even if the object is moving in two directions (ie. going down a ramp, in which it would be moving up and down and left and right), we would only focus on one of those in this subject, that is either left and right or up and down

The Terminology and Their Relation

x(t)= position v(t)= velocity, v(t)=d/dx of x(t) a(t)= acceleration, a(t)= d/dx of a(t)

Above you can see the common equation notation for each component of a motion question, with velocity being the derivative of the position function, and acceleration being the derivative of the velocity function.

Position is the object’s position at any point in time. Velocity is the objects speed (in different directions), at any point in time. Acceleration is the rate at which velocity is changing, in different directions, at any point in time.

If velocity and acceleration are opposite signs, that is, velocity is positive and acceleration is negative (and vice versa), then the particle is slowing down, because acceleration is working against velocity. If velocity and acceleration are the same signs, that means the particle is speeding up, as acceleration is working with velocity.

Your Role and Work

If given an x(t) function, you can derive it to find everything else, and then just plug in for points t. That is the easiest method given to you, as long as the function for x(t) isn’t too hard.

If given a v(t) function you might have to use an integral, unless you haven’t been taught that yet. Ah, a good time to remind you to always check what type of function they give you, don’t take a velocity function and use it as you would a position function. With a velocity function you could be asked a certain speed at a time, or a certain acceleration, and you’d have to work through that to proceed.

Graphs

If given a graph of any of these functions, you’ll probably be asked to translate it into the other function’s graphs. You can do this by knowing that the slope of the x graph is (let v(t)=y) the y at any given point t for velocity. That is also the same for v, that is the slope of v is the y at any given point for acceleration. Here is a good video on the subject.

If velocity is negative, position is decreasing, if velocity is positive, the position is increasing. Don’t mix these up. Even if velocity is decreasing, as long as it’s above the x-axis, position continues to increase. It’s an easy trap to fall into so be careful.

Now, here’s a bit more of a tricky one. Say you’re given the v(t) function, and are then asked to find the displacement. At this point, you’re a bit confused, but worry not, there is a simple way to do this. To find the displacement (that is how much the position has changed in a given time interval), just look at the are under the cover in that time interval. So say, if v(t)=2, and we were asked for the diplacement on 0<t<2, then the displacement in that time interval would be 4. because the area under the curve in that graph is 4 (sketch it out). If told to use this method the area under the curve will be possible to use.

More

Now, what if they ask you for position instead of displacement? Usually if this is done, it’s because you haven’t been given the x(t) function, so, how could you figure this out when you only have the displacement? Very easy! Just click on our code link here and join Rai- just kidding, it is quite simple though, take the individual x(t) position they gave you, and then add the displacement to it, and you get the position.

If told to found distance, that is the absolute value of displacement. Displacement is a vector therefore it can go either positive or negative, but when finding distance all negative’s turn positive. Please, begging, do not mix up displacement and distance and position, they are all distinctly different things that you will have to internalize before moving onto different things.

Conclusion

Well that’s one dimensional motion, complicated huh? Well it gets worse from here! No, but really, try to internalize all the relationships and defintions and all, and it’ll be a breeze from there, usually the questions given aren’t too hard in these cases