A particle accelerates
uniformly. At the time 𝑡 equals 0.0 seconds,
the particle has a velocity 𝑣 equals 14𝑖 plus 22𝑗 meters per second. At 𝑡 equals 3.8 seconds, the
particle has a velocity 𝑣 equals 0.0𝑖 plus 11𝑗 meters per second. What is the acceleration of the
Since we’re solving for the
acceleration of the particle over a time interval from 𝑡 equals 0.0 to 3.8 seconds,
we know that we’re solving for an average acceleration. To begin on our solution, we can
recall the mathematical relationship for average acceleration. The average acceleration of an
object is equal to its final velocity minus its initial velocity divided by the time
interval over which that velocity change happens.
In our case, we could write that
our initial time 𝑡 sub 𝑖 is 0.0 seconds. Our initial velocity is 14𝑖 plus
22𝑗 meters per second. And our final time is 3.8 seconds,
and our final velocity is 0.0𝑖 plus 11𝑗 meters per second. If we calculate the difference 𝑣
sub 𝑓 minus 𝑣 sub 𝑖, we find, as we treat these vectors separately by their 𝑖
and 𝑗 components, that we end up with a total vector of negative 14𝑖 minus 11𝑗
meters per second.
So, when we go to calculate our
average acceleration, we have this vector as our numerator divided by our time
difference of 𝑡 sub 𝑓, 3.8 seconds, minus 𝑡 sub 𝑖, 0.0 seconds. Our total time interval then is 3.8
seconds. And when we calculate this
fraction, we find a result of negative 3.7𝑖 minus 2.9𝑗 meters per second
squared. That’s the acceleration experienced
by the particle over this time interval.