calculus
Ok here is the problem:
A ball is launched vertically from the ground with an initial velocity of 96ft/sec.
How can I find the equation or at least how can I start this problem?
Here are some questions related to the problem.
a. How long does it take the ball to rise to the maximum height?
b. What is the maximum height?
c. How long does it take the ball to reach halve the initial velocity?
d. What is the height of the ball when its velocity is half the initial velocity?
I think you need some physical reasoning here. Once you fire your bullet vertically, it's only slowed by gravity until it's velocity is zero and then it starts to fall back. So, in absolute value, velocity is maximum in the moment of firing and when it returns to the surface. If you think about the vertical direction as positive, then the maximum occurs precisely in the firing moment. The reason why the derivative test doesn't work here is the following. Mathematically, it's stated as the proposition:
"If $f$ has a relative maximum in an open interval, then $f'$ is zero in this point"
The problem here is that velocity is a decreasing linear function of time, say $v(t) = v_{0} - gt$, where $v_{0}$ is the firing velocity and $g$ is gravity. Also, you're working on a closed interval $[0, t_{max} ]$, where $t_{max}$ is the instant corresponding to maximum height, calculated as above by you. Notice that $v(t)$ has no relative minimum in $(0, t_{max} )$, and that's why the test doesn't work (indeed, it's derivative is a constant). On the other hand, Weierstrass Theorem assures you that the continuous function $v(t)$ attains a maximum in the compact interval $[0, t_{max} ]$. Since you know it's not in the interior, it must be one end-point, and it's clear that it's actually $0$.
If you know the maximum height, the answer is really simple to find - we can directly work from there, knowing that the velocity at maximum height is $0 \frac m s$ because it is, essentially, not moving as it turns from upwards motion to downwards free-fall.
Note that there are two approaches to finding this solution - one, more basic and perhaps easier to understand (but longer and more complex), the kinematic approach, using basic ideas of velocity, acceleration, and displacement - and the other, although more advanced and, much simpler - the energy approach. I'll start with the basic solution using the kinematic approach.
We know that, for any object under constant acceleration, the velocity at any point in time can be modeled by
$$v = v_0 + at$$
As previously stated, because the ball falls starting from its maximum height, the initial velocity is $0$. Therefore, the equation becomes
$$v = at$$
where $v_0$ is the initial velocity, $a$ is acceleration, and $t$ is the time elapsed. For most free fall cases, $a = -g$ (where $g$ is gravitational acceleration) as gravitational acceleration is downwards. However, in this particular case, we already know the maximum height and only need to examine the part of the trajectory that is downwards - therefore, if we simply "rotate our axes," setting the direction of motion as the positive direction, we can set $a = g$ because gravity would be working in the same direction of our current motion. Therefore, we can modify the equation to become
$$v = gt$$
Now all that's left is finding the time elapsed to reach the ground. We know, first of all, that we can find the displacement of an object under constant acceleration by averaging the initial and final velocities and multiplying that by time, thus
$$\Delta x = \frac {t(v_0 + v)} 2$$
Again, here, we know $v_0 = 0$, so the equation becomes
$$\Delta x = \frac {t(v)} 2$$
From here, we can solve for $t$ and then substitute our solution for $t$ into the previous equation to solve for $v$.
\begin{align*} \Delta x &= \frac {t(v)} 2 \quad && \\ \frac {2\Delta x} v &= t \quad && \text{Multiply both sides by $\frac 2 v$} \\ t &= \frac {2\Delta x} v \quad && \text{Solve for $t$} \end{align*}
Then, we substitute that into the previous equation:
\begin{align*} v &= gt \quad && \text{Previous equation} \\ v &= g(\frac {2\Delta x} v) \quad && \text{Substitute for $t$} \\ v &= \frac {2g\Delta x} v \quad && \text{Simplify fractions} \\ v^2 &= 2g\Delta x \quad && \text{Multiply both sides by $v$} \\ v &= \sqrt{2g\Delta x} \quad && \text{Square root both sides} \end{align*}
Then, we can plug in everything we already know to solve for $v$. Because of personal preferences ;), I'll do it in SI units:
\begin{align*} v &= \sqrt{2 \cdot 9.81 \frac m {s^2} \cdot 40.23 m} \\ &\approx 28.04 \frac m s \\ &\approx 91.99 \frac {ft} s \text{ or } 92 \frac {ft} s \end{align*}
Assuming a system consisting of just the ground (gravitational force) and the ball, we can apply the Law of Conservation of Energy, which states that, in any closed system, the net energy (kinetic + potential) is constant. Here, we need only to find the energy at the two endpoints of motion and equate the two.
As previously stated, the velocity is 0 at the peak of motion - therefore, the only energy present would be potential energy. We know that gravitational potential energy is given by $mgh$, where $m$ is the mass of the ball, $g$ is the gravitational acceleration, and $h$ is the height. Thus
$$E = mgh$$
Here, when the ball is at the ground, the height $h$ is essentially $0$ (except that tiny (negligible) bit, as it isn't quite in contact with the ground yet), and all that needs to be measured is kinetic energy. We know that kinetic energy is given by $\frac 1 2 mv^2$, therefore
$$E = \frac 1 2 mv^2$$
Now, we can easily equate the energy representations at different points in the ball's trajectory at the two endpoints and solve for $v$.
\begin{align*} mgh &= \frac 1 2 mv^2 \quad && \text{Equate the two energy representations} \\ gh &= \frac 1 2 v^2 \quad && \text{Divide both sides by $m$} \\ 2gh &= v^2 \quad && \text{Multiply both sides by 2} \\ \sqrt{2gh} &= v \quad && \text{Square root both sides} \\ v &= \sqrt{2gh} \quad && \text{Solve for $v$} \end{align*}
Unsurprisingly, this yields the exact same result as it would with the kinematics approach - but is a much shorter and more concise solution - the only downside to this solution is that it requires a deeper understanding of Physics concepts of work and energy.
Best Answer
Depending on the value you want to find, the equations may vary - of course, you can always manipulate one to become another through substitution, but I'll try to make it simple; for your question, I'll start with two basic equations.
To start with, we know the gravitational acceleration is $g = 9.81 \frac m {s^2}$ or $g = 32.19 \frac m {s^2}$, and from basic kinematics, the velocity at any time is modeled by $v_f = v_0 + at$ where $v_f$ is the final velocity, $v_0$ is the initial velocity, $a$ is the acceleration, and $t$ is the time elapsed.
Equation 1. Change in Velocity
In our case, $a = -g$ as acceleration is downward (characteristics of free fall), so we can substitute in our starting equation to get
$$v_f = v_0 - gt$$
Equation 2. Change in Position (Displacement)
Also, we know the displacement (net distance traveled) can be found by averaging the initial velocity and the final velocity and then multiplying by the time elapsed, thus
$$\Delta x = \frac {t(v_0 + v_f)} 2$$
And we'll work from there.
A. How long does it take the ball to rise to the maximum height?
From common sense, first of all, you can realize that, at maximum height, the ball's velocity is $v_f = 0 \frac m s$ (it's essentially not moving, as its maximum height is right where the ball stops moving up and starts moving down. Before we substitute quickly: I think it's good practice to solve for $t$ symbolically first, so let's do that:
\begin{align*} v_f &= v_0 - gt \quad && \text{Start with first equation} \\ v_f + gt &= v_0 \quad && \text{Add $gt$ to both sides} \\ gt &= v_0 - v_f \quad && \text{Subtract $v_f$ from both sides} \\ t &= \frac {v_0 - v_f} g \quad && \text{Divide both sides by $g$ (4)} \\ t &= \frac {v_0} g \quad && \text{Here, we know $v_f = 0$} \end{align*}
Then, you can easily find $t$ by substituting $v_0$, $v_f$, and $g$ - all of which are given.
B. What is the maximum height?
For this case, we can use the second equation and solve for $\Delta x$ after substituting for $v_0$ and $v_f$, which are given, and $t$ which we can get from the equation above. Substituting and simplifying, we get:
\begin{align*} \Delta x &= \frac {t(v_0 + v_f)} 2 \quad && \text{Start with second equation} \\ \Delta x &= \frac {(v_0 - v_f)(v_0 + v_f)} {2g} \quad && \text{Substitute for $t$ and simplify} \\ \Delta x &= \frac {v_0^2 - v_f^2} {2g} \quad && \text{"Expand" polynomial (3)} \\ \Delta x &= \frac {v_0^2} {2g} \quad && \text{Here, we know $v_f = 0$} \end{align*}
C. How long does it take the ball to reach half the initial velocity?
Although seemingly trickier, this is really not much harder at all than Question A - in fact, much of the work has already been done in that question; we can directly start from the fourth step of the solution to Question A (labeled in parentheses), where we have solved for $t$. Here, $v_f = \frac {v_0} 2$.
\begin{align*} t &= \frac {v_0 - v_f} g \quad && \text{Start with equation in step 4 of A} \\ t &= \frac {v_0 - \frac {v_0} 2} g \quad && \text{Substitute for $v_f$} \\ t &= \frac {\frac {v_0} 2} g \quad && \text{Simplify} \\ t &= \frac {v_0} {2g} \quad && \text{More simplify} \end{align*}
D. What is the height of the ball when its velocity is half the initial velocity?
Again, this is just like Question B. Therefore, we can start with the equation in the third step of the solution to Question B.
\begin{align*} \Delta x &= \frac {v_0^2 - v_f^2} {2g} \quad && \text{Start with equation in step 3 of B} \\ \Delta x &= \frac {v_0^2 - (\frac {v_0} 2)^2} {2g} \quad && \text{Substitute for $v_f$} \\ \Delta x &= \frac {v_0^2 - \frac {v_0^2} 4} {2g} \quad && \text{Simplify} \\ \Delta x &= \frac {\frac {3v_0^2} 4} {2g} \quad && \text{More simplify} \\ \Delta x &= \frac {3v_0^2} {8g} \quad && \text{Even more simplify} \end{align*}
And there we have it - the solutions to each of the four problems.