in this problem we're doing one-dimensional kinematics but we have two moving objects so
the diagram gets a little bit complicated at t equals zero car a is going through the origin
at 32 meters per second and car a moves with constant velocity so it's a very simple motion
it just keeps going at 32 meters per second at the same moment in time car b starts from rest
and it accelerates in the positive direction with a constant rate of 8 meters per second squared so
car b starts from rest car a blows right by it and then car b just speeds up and speeds up and
speeds up and eventually goes zooming past car a and we're trying to find the position where that
occurs we're trying to find the time where that occurs and we're trying to find the speed of car b
when it goes zooming past car a so the first step in a problem like this is to be able to
mathematically describe the position of each car as a function of time and this just goes
back to our standard kinematics equations so our first sort of fundamental kinematics equation
is x equals x naught plus v naught t plus one half a t squared
each of these cars has a constant acceleration so they each obey an equation like this
describing their position as a function of time for part d we're going to need the second
fundamental kinematics equation so i may as well write it down right now v equals v naught plus a
t so in part a we're asked for the position as a function of time for each car and it's really
just this first equation of motion that i wrote down each of them has an initial position of zero
and if i start working on car a it has an initial velocity of 32 meters per second notice i skipped
the x naught term because it's located at zero when t is zero and car a has an
acceleration of zero as well so we're done that's the position as a function of time
car b is a little more complicated it also starts at x equals zero so that term is gone
um it has an initial velocity of zero so that term is gone but it has a non-zero acceleration so that
term survives the acceleration is eight so i get one half times eight times t squared which i can
simplify then to four t squared so there's my two functions that describe the position of these cars
as a function of time that allows me to answer part b compute the time at which car b passes
car a and the key to this is to say well when car b passes car a the two cars are in the same place
at the same time so i just set those two position functions equal to each other and i ask
what time is it so i get 32t is equal to 4t squared i can cancel a t out of this but i want
to point out that when i do that i'm eliminating one of the solutions to this quadratic equation
t equals 0 is a solution and the reason why is there's two different times where the two cars
are in the same place one of them of course was at t equals zero that's where the problem started
and then the one i really am interested in is the other one so we can go ahead and cancel out
that extraneous solution dividing by four i end up with t equals eight seconds in part c i'm asked to
get the position at which car b passes car a and you can plug into either one of these position
functions to get that done and it's actually wise to do both of them just to make sure the
answers agree so if i look at the position of car a when t is 8 seconds i get 32 times 8 and that's
256 meters to the right of the origin when i look at car b i'd better get the same answer
and that's also 256 meters to the right of the origin
so we verify these two cars really are in the same place when t is equal to 8 seconds so that's good
finally we're asked to compute the speed of car b when it passes car a
so this is where we use the second basic kinematics formula v equals v naught plus a t
and i subscripted all those quantities with a b because that's the car we're talking about
and i'm interested in its final velocity when t is equal to eight and the initial velocity was zero
the acceleration is 8 meters per second squared
the time it was accelerating for is 8 seconds and i get 64 meters per second
if you find the physics content on zack's lab helpful click on the zackslab logo on
the right to browse playlists and subscribe to the channel i produce over 100 new videos per
month and subscribing is the easiest way to find new content thanks for watching