Video Transcript
A soldier jumped out of a plane
with a parachute. After he had opened his parachute,
the resistance to his movement was directly proportional to the cube of his
speed. When his speed was 19 kilometers
per hour, the resistance to his motion was a twenty-seventh of the combined weight
of him and his parachute. Determine the maximum speed of his
descent.
Let’s sketch a little diagram. Here is our soldier plummeting
through the sky. The downwards force of his weight
is what’s causing him to do so. Then, we have a resistance to the
motion of the parachute, that’s 𝑅. And it acts in the opposite
direction. We’re told that the resistance is
directly proportional to the cube of his speed. So, let’s let his speed be equal to
𝑣, and that’s kilometers per hour. For this to be the case, 𝑅 must be
equal to 𝑘 times 𝑣 cubed, where 𝑘 is known as the constant of
proportionality. Let’s find an expression for 𝑘 by
using the rest of the information in this question.
When his speed was 19, the
resistance was a twenty-seventh of the combined weight of him and his parachute. So, when 𝑣 is 19, 𝑅 is one
twenty-seventh times 𝑤. So, we can say that one
twenty-seventh times 𝑤 is 𝑘 times 19 cubed. This means 𝑘 is a twenty-seventh
𝑤 divided by 19 cubed. Now, that’s equivalent to saying 𝑤
over 27 times 19 cubed. Now, we’re not going to work this
out and we’ll see why in a moment. We can, therefore, substitute this
back into our earlier equation and say 𝑅 is 𝑤𝑣 cubed over 27 times 19 cubed.
Now, we’re told that at some point
the soldier reaches a maximum speed. At this point, the speed will
remain unchanged. Now, for the velocity to remain
uniform, Newton’s first law of motion says that the sum of all of the forces in this
direction must be equal to zero. Now, let’s take downwards to be the
positive direction. We can say that the weight minus
the reaction force is the sum of these forces. So, 𝑤 minus 𝑅 is equal to
zero. Then, adding 𝑅 to both sides, we
get 𝑤 is equal to 𝑅.
But let’s replace 𝑅 with the
earlier expression in terms of 𝑤 and 𝑣. And since 𝑤 is not equal to zero,
we’re able to divide both sides of this equation by 𝑤. So, one is 𝑣 cubed over 27 times
19 cubed. We’re then going to multiply both
sides by 27 times 19 cubed. And finally, we’re going to find
the cube root of both sides. The cube root of 27 is three, and
the cube root of 19 cubed is 19. So, we find three times 19 is equal
to 𝑣, but three times 19 is 57. So, 𝑣 is equal to 57 and that’s
kilometers per hour.
