in this video we're talking about a very unusual projectile motion
problem so we're given the launch speed and launch angle for a projectile but what we're asked for
is the average velocity vector from t equals 0.5 seconds to t equals 1.5 seconds so this problem
is meant to get into the formal vector representation of position and velocity
so what we want to start with here is to represent the position of the projectile
as a function of time and then we should be able to compute that average velocity vector
and just as a quick reminder a position vector if i put my origin right here at the launch point
a position vector points to the location of the object at each moment in time as it's going
through this projectile flight so at 0.5 seconds it's pointing up here at 1.5 seconds it's pointing
over here and everywhere in between that position vector is just tracing the projectile through its
flight that vector is rotating and growing over time to trace this trajectory out so how do we
find that vector that points from the origin to the position of the object at each moment in time
we can write this thing down in general as an x coordinate as a function of time times i hat
plus a y coordinate as a function of time times j hat but we know the equations that describe
projectile motion and that will give us the x of t and the y of t so x of t for a projectile
is going to be x naught plus v naught x times t
that's constant velocity motion in the horizontal direction and we know y of t for a projectile
that's y naught plus v naught y t minus one half g t squared where g is the magnitude of the
acceleration of gravity in our initial position here we put the origin right at the initial
position so x naught and y naught are both zero then i need to get my initial velocity components
and then i'll have my general position vector describing the position of this projectile
as a function of time and then we can get into this average velocity thing
so let's get the initial velocity components first v naught x is going to be a 10 cosine 65
into three sig figs that gives me 4.23 meters per second v naught y is a 10 sine 65
and that gives me 9.06 meters per second and now i can write down my general position vector
and that's v not x t for the x coordinate so 4.23 t
i hat and then for the y coordinate v not y is 9.06 so 9.06 t
minus half of g which is 4.9 times t squared times j hat now we need a reminder of what
the average velocity vector is and that's a change in position divided by a change in time
we could also call that change in position the displacement of the object on this time
interval that's actually a vector pointing from the location of this object at t1 to the location
of the object at t2 and that's given by r of 1.5 seconds minus r of 0.5 seconds
and then we divide that by the time it took to get from point number one to point number two
which is exactly one second that's t equals 1.5
for the final time and t equals 0.5 for the initial time
and this average velocity vector points in the exact same direction as the displacement vector
but it's been scaled by a factor of delta t and now all we have to do is plug in the numbers to
the position vector combine like terms for the unit vectors and clean it up and we'll have our
average velocity so first r of 1.5 so i sub in 1.5 for t when i do that on the i hat term
i get about 6.35 i hat and then on the j hat term i get about 2.57 j hat
then i subtract the position vector that i get by subbing in 0.5 for t that gives me a 2.11 i hat
plus a 3.31 j hat and then my time interval in the denominator is just one second when i run
the numbers here i get 4.24 i hat minus 0.74 j hat and to state my final answer i really should
have units on these so these coefficients must have units of meters per second so on average
during this time interval my projectile was moving 4.24 meters per second to the right
and negative 0.74 meters per second downward and we're done if you enjoyed this video or at
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