Understanding the Use of the Long Jump Formula R=9.21m and Its Derivation

[rudransh verma]

rudransh verma said:
I am questioning the existence of this formula. When you can do it via the obvious way by finding t why have we derived a new formula?

As I posted, we first have to find t without using θ=π/4. We can do that by writing the two usual equations:
y=vsin⁡(θ)t−12gt2, x=vcos⁡(θ)t
Substituting y=0 to find the time when it lands:
2vsin⁡(θ)t=gt2, 2vsin⁡(θ)=gt
Plugging that into the x equation:
x=vcos⁡(θ)2vsin⁡(θ)/g=v2sin⁡(2θ)/g
Now we can see x is maximised by θ=π/4.

So if I understood it correctly 1) we cannot find t because we don’t know at what $\theta$ t is such that R is max provided t is a function of $\theta$.
2) t is a function of $\theta$ shown via eqn $0={v_0}\sin\theta t-\frac12gt^2$

 

[rudransh verma]

It is just a hop, a skip and a jump from there to concluding that the angle that maximizes R is 45 degrees.

This is the standard crank-and-grind approach that you should try when attacking almost any simple optimization problem: 1. Obtain or manipulate a formula for the value-to-be-optimized so that there is only one free variable and everything else is a constant. 2. Solve for the value that makes the first derivative of this function zero. 3. Evaluate the function at that value.

$$R={v_0}\cos\theta t$$
Put the value of t we obtained from
$0={v_0}\sin\theta t-\frac12gt^2$
$$R=\frac{{v_0}^2\sin2\theta}{g}$$
Differtiating R with respect to $\theta$ and equating to zero we get $\cos2\theta=0$. $\theta=45$ . Putting in original eqn of R we get R maximised.

 

[haruspex]

The differentiation is unnecessary since we know the max value of sine is 1.

 

[rudransh verma]

The differentiation is unnecessary since we know the max value of sine is 1.

By the way what is the purpose of differentiating(I guess to find instantaneous rate) but why have we equated it to zero?
All we were doing was trying to maximise R.

 

[jbriggs444]

By the way why are we differentiating and why have we equated it to zero?

Recall our goal. We are trying to find the maximum range $R$ for a projectile launched at a known velocity $v_0$ at an unknown angle $\theta$ when the landing position is at the same elevation as the launching position. We are free to pick $\theta$ in order to maximize $R$.

We have come up with a formula that gives the achieved range, call it $R(\theta)$ for any given angle. That formula is:$$R(\theta)=\frac{{v_0}^2 \sin 2\theta}{g}$$This formula has been given many times already in this thread.

When one has an arbitrary function $f(x)$ defined for $x$ in some interval, there only a few places where a maximum or a minimum can be.

The maximum can be at one of the end points of the interval. For example, the function $f(x)=x$ on the closed interval [0,1] has its maximum at the right hand end point and its minimum at the left hand end point.

The maximum can be at a place where the function is discontinuous or where it is not differentiable. For example, the function $f(x) = |x|$ on the closed interval [-1,1] has its minimum at the bottom of the V that the graph traces out. The first derivative there is undefined.

The maximum can be at a place where the first derivative of the function is zero. For instance, the function $f(x)=\sqrt{1-x^2}$ on the closed interval [-1,1] has its maximum at the peak of its circular arc where the first derivative is zero.

One crank-and-grind approach to optimization problems is, accordingly, to solve for places where the first derivative is zero. The maximum may be at such a place.

As @haruspex points out, once one has a function of the form $$\text{some positive constant} \times \sin 2\theta$$there is no need to go the trouble of differentiating. It is clear by inspection where the maximum is to be found. [Still, nothing stops you from differentiating and solving $\text{some positive constant} \times \cos 2\theta = 0$ for $2\theta$. Easily enough done. This yields the correct result, as it must]

If no such obvious solution is to be found and if one is being careful [as one should be], one should check at the endpoints of the interval (e.g. at $\theta = 0 \text{ degrees}$ and $\theta = 90 \text{ degrees}$), at any possible points of discontinuity and at any points where the first derivative is zero or undefined to see at which point(s) the maximum (or minimum) is actually achieved.

 

[kuruman]

Here is another way to look at it. First rewrite the range $R$ in terms of the initial velocity components, $v_{0x}$ and $v_{0y}$ this way
$$R=\frac{v_0^2 \sin(2\theta)}{g}=\frac{v_0^2 (2 \sin\theta\cos\theta)}{g}=\frac{2(v_0 \sin\theta)(v_0\cos\theta)}{g}=\frac{2v_{0x}v_{0y}}{g}=\frac{2}{g}v_{0x}v_{0y}.$$Note that you can think of the range as proportional to the area of a rectangle of base $v_{0x}$ and height $v_{0y}$, the constant of proportionality being $\frac{2}{g}$. For an arbitrary choice of base and height you have two rectangles of the same area. For example, a projectile launched with $v_{0x}= 3~\text{m/s}$ and $v_{0y}= 4~\text{m/s}$, will have the same range (but different projection angle) as a projectile launched with $v_{0x}= 4~\text{m/s}$ and $v_{0y}= 3~\text{m/s}$. Because there are two different projection angles with the same range, the range cannot be maximum (cannot have the largest value) at these angles.

However, when the interchange of $v_{0x}$ and $v_{0y}$ results in the same projection angle $\theta$, you have a maximum^* because there is only one value for the range at that angle. This happens when $v_{0x}=v_{0y}$, i.e. the horizontal and vertical components of the initial velocity are equal and the projection angle is 45°.
__________________________________________________
^* It is a maximum because the range is zero when either one of the components is zero.

 

[jbriggs444]

Because there are two different projection angles with the same range, the range cannot be maximum (cannot have the largest value) at these angles.

I like arguments from symmetry. But there is a loophole in this one.

You've successfully proved that if the function has a unique global maximum then that maximum must be achieved when $v_{0x}= v_{0y}$. There is nothing that prevents a function $f(x)$ valid on the range [0, $\pi$] and satisfying the property that $f(x) = f(\pi-x)$ from having a non-unique global maximum in that range and then not achieving its global maximum at $x=\frac{\pi}{2}$

 

[kuruman]

I like arguments from symmetry. But there is a loophole in this one.

You've successfully proved that if the function has a unique global maximum then that maximum must be achieved when $v_{0x}= v_{0y}$. There is nothing that prevents a function $f(x)$ valid on the range [0, $\pi$] and satisfying the property that $f(x) = f(\pi-x)$ from having a non-unique global maximum in that range and then not achieving its global maximum at $x=\frac{\pi}{2}$

I am not sure I understand the loophole. If I choose a projection angle conventionally in the first quadrant of the unit circle such that $v_{0x}\neq v_{0y}$, then the product of the two, and hence the range, is doubly degenerate. This means that the range for this particular choice of angle cannot be an extremum. All angles in the first quadrant have a counterpart that results in the same range except the 45° angle which is its own counterpart. Thus, the range is at an extremum at 45°. The extremum is a maximum because the range is zero at $\theta =0$ and at $\theta = \pi/2$ and has positive values in-between.

 

[jbriggs444]

This means that the range for this particular choice of angle cannot be an extremum.

The symmetry argument guarantees that the range at some off-center angle cannot be greater than at all other angles. But that says nothing about whether it can be greater than or equal to the range at all other angles.

For instance, $f(x)=\sin 6x$ over the range $[0,\frac{\pi}{2}]$ has the symmetry properties that you are invoking. $f(x) = f(\pi-x)$. However, it has local and global maxima at $\frac{\pi}{12}$ and $\frac{5\pi}{12}$ and a global minimum at $\frac{\pi}{4}$. (15 degrees and 75 degrees for the twin maxima and 45 degrees for the global minimum)

 

[kuruman]

I think we are talking past each other. I am not invoking a general symmetry principle. I have an area which I will write as $A=v_{0x}\sqrt{v_0^2-v_{0x}^2}$ and maximize as $v_{0x}$ is varied from zero to $v_0$. It has only one global maximum when the two factors are equal.

 

[jbriggs444]

I think we are talking past each other. I am not invoking a general symmetry principle. I have an area which I will write as $A=v_{0x}\sqrt{v_0^2-v_{0x}^2}$ and maximize as $v_{0x}$ is varied from zero to $v_0$. It has only one global maximum when the two factors are equal.

So you are making use of the well known fact that the area of a rectangle of fixed perimeter is maximized when the rectangle is a square. Sure, that works.

 

[rudransh verma]

The maximum can be at one of the end points of the interval. For example, the function f(x)=x on the closed interval [0,1] has its maximum at the right hand end point and its minimum at the left hand end point.

I suppose the maximum of this function is at 1 as 1. Seems obvious.

The maximum can be at a place where the function is discontinuous or where it is not differentiable. For example, the function f(x)=|x| on the closed interval [-1,1] has its minimum at the bottom of the V that the graph traces out. The first derivative there is undefined.

The max here I suppose is at both 1 and -1.

The maximum can be at a place where the first derivative of the function is zero.

Is it a thumb rule?Is it a nature of graphs? I have not studied much about derivatives except how to do basic differentiation.

 

[haruspex]

Is it a thumb rule?Is it a nature of graphs?

Assuming the curve is smooth, if the gradient is >0 then the function is increasing, so hasn't reached the maximum yet; if the gradient is negative the function is decreasing, so has passed the maximum. To be at a local maximum (or minimum) the gradient must be zero.

 

[rudransh verma]

Assuming the curve is smooth, if the gradient is >0 then the function is increasing, so hasn't reached the maximum yet; if the gradient is negative the function is decreasing, so has passed the maximum. To be at a local maximum (or minimum) the gradient must be zero.

You mean when there is no rate + or - then the function is either at its maximum or minimum. The function is increasing/decreasing at an instant if the derivative is not equal to zero.

How do you know all this? Is it university grade stuff? All we are taught in 10+2 level is how to do differentiation and some differential equations.

 

[haruspex]

How do you know all this?

My post was intended as self evident.

 

[rudransh verma]

My post was intended as self evident.

It’s not obvious if that you mean

 

[jbriggs444]

You mean when there is no rate + or - then the function is either at its maximum or minimum.

Be careful. The implication is the other way around.

If the rate is + or - then the function is not at its maximum or minimum.
If the rate is 0 then the function might be at a maximum or minimum.

Take the example of $f(x) = x^3$. The first derivative of this function is zero at $x=0$ but there is no maximum or minimum there.

The function is increasing/decreasing at an instant if the derivative is not equal to zero.

Yes. This statement is correct.

If the derivative is non-zero, the function is increasing or decreasing and, therefore, is not at a maximum or minimum. Intuitively, it is like climbing a hill. If the ground is sloping, you are not at the top yet.

Here is an on-line lesson.