# Max Time to Reach Apex: Exploring the Surprising Involvement of Pi"

• gamesguru
In summary, the conversation discusses the maximum time it takes for an object thrown upward to reach its highest point, which is represented by the equation t_{max}=\frac{\pi}{2}\sqrt{\frac{m}{gk}}, where k=\frac{1}{2}\rho A C_d is the drag constant. The conversation also explores the relationship between velocity and time using the drag formula and the inverse tangent function. Finally, it is noted that if the object is launched at a speed greater than escape velocity, it will not reach its apex and will not come back down.

#### gamesguru

The basic point to this is to show that, no matter how fast an object is thrown up (assuming that $g$ is relatively constant), that there is a maximum time it will take to reach its highest point ($v=0$), more interesting however is that exact number which strangely involves pi:
$$t_{max}=\frac{\pi}{2}\sqrt{\frac{m}{gk}}$$.
Where $k=\frac{1}{2}\rho A C_d$ is the drag constant.

We begin by giving the object an upward initial velocity $v_0$.
Using the drag formula,
$$F=ma=-(mg+kv^2).$$.
Simplifying,
$$v'=\frac{dv}{dt}=-(g+cv^2)$$.
Where $c=k/m$.
Separating variables,
$$\frac{dv}{g+cv^2}=-dt$$.
The limits can be found by imagining the velocity going from $v_0$ to $v_f$, and time going from $0$ to $t$,
$$\int^{v_f}_{v_0}\frac{dv}{g+cv^2}=-\int^t_0 dt$$
After some substitution, we arrive, not-surprisingly, at something involving the inverse tangent:
$$-t=\frac{\tan^{-1}v_f\sqrt{\frac{c}{g}}-\tan^{-1}v_0\sqrt{\frac{c}{g}}}{\sqrt{cg}}$$.
At the apex,$v=0$, and since $\tan^{-1}0=0$, we find,
$$t_{max}=\frac{\tan^{-1}v_0\sqrt{\frac{c}{g}}}{\sqrt{cg}}$$
Since it is physically observable that objects thrown upwards more quickly take longer to reach their apex, we take the limit of $t_{max}$ as $v_0\rightarrow\infty$.
$$\lim_{v_0\rightarrow \infty}\frac{\tan^{-1}v_0\sqrt{\frac{c}{g}}}{\sqrt{cg}}=\frac{\pi/2}{\sqrt{cg}}$$.
Substituting back in, we arrive at the final equation for the maximum time it will take an object fired to reach its apex:
$$t_{max}=\frac{\pi}{2}\sqrt{\frac{2m}{g\rho A C_d}}$$.

I didn't understand any of those characters except for the '=', '+' & '-' signs. If the thing is launched at greater than escape speed, however, how can there be an apex? It ain't coming back down.

Danger said:
I didn't understand any of those characters except for the '=', '+' & '-' signs. If the thing is launched at greater than escape speed, however, how can there be an apex? It ain't coming back down.
"(assuming that $g$ is relatively constant)"
This means that g=9.8 m/s^2 and stays sufficiently close to 9.8. But, you're right, if it is launched fast enough, it won't come back down.

## 1. What is the significance of Pi in the concept of "Max Time to Reach Apex"?

The involvement of Pi in the concept of "Max Time to Reach Apex" is due to its mathematical properties and its relationship to circles and curves. In this concept, Pi is used to calculate the trajectory of an object and determine the maximum time it takes for the object to reach its apex or highest point.

## 2. How does Pi affect the time it takes for an object to reach its apex?

Pi affects the time it takes for an object to reach its apex by influencing the shape of the object's trajectory. The value of Pi determines the curvature of the trajectory, which ultimately affects the time it takes for the object to reach its highest point.

## 3. Can Pi be used to calculate the maximum time to reach apex for any object?

Yes, Pi can be used to calculate the maximum time to reach apex for any object as long as the object follows a curved trajectory. This can include objects such as projectiles, roller coasters, and even planets in orbit.

## 4. How accurate is the use of Pi in calculating max time to reach apex?

The use of Pi in calculating max time to reach apex is highly accurate. Pi is an irrational number with infinite decimal places, making it extremely precise in mathematical calculations. However, the accuracy also depends on the accuracy of other variables in the calculation, such as the initial velocity and angle of the object.

## 5. Are there any real-world applications of "Max Time to Reach Apex" and Pi?

Yes, there are numerous real-world applications of "Max Time to Reach Apex" and Pi. One example is in sports such as baseball, where the trajectory of a ball can be calculated using Pi to determine the maximum time it takes for the ball to reach its highest point. Pi is also used in engineering and physics to calculate the trajectory of objects in motion, such as rockets and satellites.