in this video we're looking at what happens to the motion of an object depending on the relative
alignment of the acceleration and velocity vectors so on the left we have the one-dimensional cases
and on the right we have the two-dimensional cases just to generalize things
and of course we start with the one-dimensional cases and in the first picture i have a cart
that's moving to the right and it's also accelerating to the right and i'm trying to figure
out what happens to the motion of this thing and just as a reminder the average acceleration so
we're just considering finite chunks of time here the average acceleration is going to be the change
in the velocity divided by the change in the time now in one dimension we normally
deal with direction by using plus and minus signs but these are technically vectors and
so what happens in a small chunk of time is that acceleration causes the velocity to change and i
can just solve for delta v here delta v is my acceleration vector multiplied by a change in
time delta t and so that vector on the right hand side has simply been scaled by a factor
of delta t and that can't change its direction so my delta v must point in the same direction as a
and there's my little delta v and what i'm doing here is taking the change in
velocity and vector adding it to the original velocity so i arrange them head to tail and
then i draw a new velocity vector from the tail of the first to the head of the second
that's v plus delta v and i can see that that's longer than the original velocity in other words
our conclusion here is if the acceleration points in the same direction as the velocity then the
speed goes up so this thing is speeding up in the second case i have an acceleration vector opposing
the direction of the velocity vector and we do all the same stuff i want to be able to solve
for delta v as a times delta t and then i realize the change in my velocity here is going to point
in the same direction as a which is to the left and that's opposite the direction of the original
velocity vector and when i vector add these two things so i'm going to pick up my delta v
and put the tail of it on the head of v then the vector pointing from the tail
of v to the head of my little delta v is going to be shorter than the original v and i discover that
my object is slowing down so whenever the acceleration vector points opposite to the
velocity vector you're looking at an object that's slowing down so what happens when we
extend this to two dimensions in the pictures on the right i have sort of three different cases
of relationships between acceleration vectors and velocity vectors in a two-dimensional space
and the key to understanding this is to break that acceleration vector into components
so in the first one there's a bit of a component pointing in the same direction
as the velocity vector and we call that a parallel it's the parallel component my acceleration vector
and there's a component pointing perpendicular to the velocity vector and it's still true that
the change in my velocity vector is given by my acceleration vector times delta t
but if i write my acceleration vector as a sum of these two components
i get a parallel times delta t plus an a perpendicular times delta t and then i see
that a parallel times delta t points in the same direction as the original vector v so
that's the first contribution to the change in the velocity vector i'll call that delta v parallel
and at this point i already see that my speed has increased and i still have to add to that
this vector component so a perpendicular times delta t well that points in the same direction as
a perpendicular and gives me a little chunk like this that's my delta v perpendicular so when
i vector add these two vectors i have a vector that's both longer but is also rotated clockwise
and the way to think of this is that parallel component of a is controlling the speed and
if that points in the same direction as v my speed goes up this perpendicular component of a
that controls the turning of the velocity vector in this case it made the velocity vector turn
clockwise let's see what happens in the second case so again i want to think about components of
this acceleration vector relative to the direction of the velocity vector so i'm decomposing with
respect to these axes and i have an a parallel that's pointing opposite the direction of v and i
have an a perpendicular that's pointing up this way again perpendicular to the direction of v
and again if i multiply by a little delta t that parallel component contributes a change in the
velocity vector that goes this way opposing the original direction that's my delta v parallel
that's going to make the speed go down and i'm vector adding three vectors here so when i
tack on that perpendicular velocity change i start from the head of the second vector here
and move in the direction of a perpendicular to get my little delta v perpendicular and the
resulting velocity vector points from the tail of the first to the head of the last
in that vector sum and there's my new velocity vector v plus delta v so in this case that
parallel component of the acceleration vector made the velocity vector shrink
so it slowed it down because that acceleration is opposing the direction of the velocity
and then the perpendicular component again caused the velocity vector to turn clockwise so what do
we do with the third case in the third case i have a very special case where the acceleration
vector is perpendicular to the velocity vector and this means there is no parallel component
to cause the speed to change so this is a critical point if you find yourself in a
situation where the acceleration is perpendicular to the velocity then the speed will stay the same
so then the only question really is which way is this thing turning and i can get that really quick
qualitatively a little change in velocity would be in the same direction as the acceleration vector
and my rotated velocity vector would look like this now you might complain that that v plus
delta v is the hypotenuse of a triangle so it must be longer than the original v but what's
happening here is i'm assuming this a continues to stay perpendicular to v as it rotates so after an
infinitesimal amount of time the a has rotated a little bit and it's always staying perpendicular
to v throughout this rotation process so v can't grow longer and by doing this quick qualitative
analysis we figured out that what we're looking at is a velocity vector that stays constant in length
but is always rotating counterclockwise and the reason this example is so important is that that's
exactly what's going to happen in uniform circular motion if you enjoyed this video or at least found
it useful check out another one by clicking one of the links on the left or click the zaxlab logo on
the right to explore dozens of physics and math playlists as always you can leave your questions
comments and requests in the comments section below and i'll get back to you within 24 hours
thanks for watching zach's lab and best of luck on your math and physics journey