Video Transcript
If a body is projected vertically
upwards with speed 𝑉 to reach maximum height ℎ, then the speed the body should be
projected by to reach height four ℎ is blank. (A) 𝑉, (B) four 𝑉, (C) two 𝑉,
(D) square root of two 𝑉.
Alright, so here we have a scenario
where a body is projected vertically upward with a speed we’ve called 𝑉. And under this influence, it
achieves a maximum height ℎ. We then imagine a scenario where
this same body is projected to a height of four ℎ. And the question is, with what
initial speed would it need to be projected to reach this height? We have these four answer options
here. And as we get started with our
answer, we can notice the fact that this body, as it moves upward, is under the
influence of only the force of gravity. Therefore, its acceleration is
uniform, and we can describe its motion using an equation of motion. The equation we’ll use is that an
object’s final velocity squared is equal to its initial velocity squared plus two
times its acceleration times its displacement.
In the case of our vertically
projected bodies, we can say that the final moment in time is the one where each one
is at its maximum height. And in each case at this point, its
velocity is zero. This means that the left-hand side
of this expression will be zero. And then, as we fill in the
right-hand side, let’s focus on this case where we have a maximum height ℎ and an
initial speed 𝑉. We would write then that zero is
equal to 𝑉 squared plus two times 𝑔, the acceleration our body undergoes,
multiplied by ℎ. It’s possible to rearrange this
equation so that it reads ℎ is equal to negative 𝑉 squared over two times 𝑔.
And in this instance, 𝑔 is equal
to negative 9.8 meters per second squared. And writing that value in without
units, we see that the negative signs in numerator and denominator cancel. So if we want a body to ascend to a
maximum height of ℎ, we need to give it an initial speed of 𝑉. And if we want a body to ascend to
a maximum height of four ℎ, where we multiply both sides of the equation by four,
then we can equivalently say that this is equal to two times 𝑉 quantity squared
divided by two times 9.8. And so, now we know what our
object’s initial velocity must be in order for it to be projected to a height of
four times ℎ. It must be twice the initial speed
𝑉.
