This lesson will deal with vertical motion problems that have initial velocities
other than 0m/s. One key thing to remember in vertical motion problems where an object
goes upward and then lands at the same level will have the same speed, but the exact
opposite velocity when it lands. For example, if a basketball player jumps off the
ground at 7m/s then, assuming he lands at the same vertical level (i. e. a hole didn't
appear in the floor), he will land with a velocity of -7m/s. It is also true that at
the very top of his jump, his velocity will be 0m/s.
Knowing this, let's do a problem together. A construction worker ascending at 6m/s
in an open elevator 42m above the ground drops a hammer.
a. How long does it take the hammer to hit the ground?
b. How fast is it moving when it hits the ground?
Since the construction worker is moving at 6m/s and he is holding the hammer, the
hammer is also currently moving at 6m/s, which means that when it is dropped its initial
velocity is 6m/s.
What we know:
a = -9.8m/s2
d = 42m
t = We need this, part a
vo = 6m/s
vf = We need this, part b
We could start with part a and use the formula d = vot + 0.5*a*t2, but
then we would have to do the quadratic equation. Instead, if we find the answer to part
b using <2ad = vf2 - vo2> first, we can use that value and the others we know to find
time without having to do the quadratic equation.
2*(-9.8m/s2)*(-42m) = vf2 - (6m/s)2
823.2m2/s2 = vf2 - 36m2/s2
859.2m2/s2 = vf2
-29.312m/s = vf
Note: The distance is negative because the hammer is going to be travelling in a
downward direction. Also remember when taking the square root of a number it can be
either positive or negative, so choose the one that works with your problem. In this
case, the hammer was going down so we needed the negative one.
Now for part a. We'll use the acceleration formula.
-9.8m/s2 = (-29.312m/s - 6m/s) / t
-9.8m/s2 = -35.312m/s / t
3.603s = t
Now, you could have completed part a first and used the quadratic equation, but
wasn't this much easier. For those of you who are just wild about the quadratic
equation, please feel free to solve this problem using it on your own time; its just
one more thing you know how to do.
Lesson 3, will put together both horizontal and vertical
motion.
