Video Transcript
A particle moving in a straight
line was accelerating at a rate of 22 centimeters per square second in the same
direction as its initial velocity. If the magnitude of its
displacement 10 seconds after it started moving was 29 meters, calculate the
magnitude of its initial velocity 𝑣 naught and its velocity 𝑣 at the end of this
period.
We’re given that the particle is
accelerating at a constant rate of 22 centimeters per square second. This means to answer this question,
we’re going to need to use the kinematic equations. These are, of course, equations of
constant acceleration. For a starting velocity 𝑣 naught,
a velocity 𝑣 after 𝑡 time units, an acceleration 𝑎, and a displacement Δ𝑥, they
are as shown.
What we do is begin by listing
everything we know about our motion. We’ve already said we know that the
acceleration is constant, and it’s 22 centimeters per square second. It’s in the same direction as its
initial velocity. Now, we don’t know its initial
velocity, but by assuming that they’re in the same direction, we can take both
acceleration and 𝑣 naught to be positive. We’re also told that the magnitude
of the displacement 10 seconds after it started moving was 29 meters. Remember, displacement can have a
direction. So, by considering just the
magnitude, we’re thinking about the distance; that’s 29 meters. Time 𝑡 is 10 seconds.
Now, the question actually asks us
to calculate the magnitude of the initial velocity and its velocity at the end of
the period. Let’s begin by calculating its
initial velocity 𝑣 naught. In this case, we’re not interested
in 𝑣, so we go through our equations and eliminate those containing 𝑣. Those are one, three, and four. Our next step would normally be to
substitute everything we know about the motion of our particle into that second
equation. We do have a little bit of a
problem though. We notice that the units for our
acceleration and our displacement are different. We need them to be the same. So, we multiply displacement by 100
and we find it’s actually equal to 2900 centimeters.
Then, substituting everything we
know into this formula, and we get 2900 equals 10𝑣 naught plus a half times 22
times 10 squared. A half times 22 times 10 squared is
1100. So, we subtract 1100 from both
sides, and we find that 1800 is equal to 10 times 𝑣 naught. Our final step is to divide through
by 10. 1800 divided by 10 is 180. Now, we’re working in
centimeters. So, our velocity, our initial
velocity 𝑣 naught, is 180 centimeters per second. We might choose to give our answer
in meters per second by dividing through by 100. And when we do, we find that 𝑣
naught is 1.8 meters per second.
We’re not quite finished. We’re still looking to calculate
its velocity 𝑣 at the end of the period. Now that we know 𝑣 naught, we can
actually use any of our equations. So, let’s use the first one. We substitute everything we know
about the motion of our particle into this formula, continuing to work in
centimeters and centimeters per second. When we do, we get 𝑣 is 180 plus
22 times 10. 22 times 10 is 220. And 180 plus 220 is 400. We’re still, of course, working in
centimeters per second. To give our answer in meters per
second, we’ll divide through by 100. And when we do, we find that the
velocity 𝑣 at the end of the motion is four meters per second.
