I .INTRODUCTION Chang-Young Lee ANumericalStudyontheLaunchingAnglefortheMaximumRangeofaProjectilewithaLinearDragForce

A Numerical Study on the Launching Angle for

the Maximum Range of a Projectile with a Linear Drag Force

Chang-Young Lee ^∗

Dongseo University, Busan 617-716

(Received 3 May 2012 : revised 28 May 2012 : accepted 30 July 2012)

In this paper, we study the launching angle that yields the maximum range of a projectile under the influence of a linear drag force due to air. The trajectory of the projectile is derived from the solution of the equation of motion, and the launching angle for the maximum range is estimated numerically in terms of a parameter that represents the ratio of the drag force to the projectile’s weight. Both an exponential and a power function are found to be inappropriate in representing the functional dependence of the desired launching angle on that parameter. Instead, polynomial fits are obtained, from which an actual procedure for implementing the maximum range is provided.

PACS numbers: 01.80.+b, 02.60.Cb

Keywords: Launching angle, Projectile motion, Maximum range, Linear drag force

I. INTRODUCTION

With ignorance of the Coriolis force whose effect is minor in most cases, the motion of a projectile thrown near the earth’s surface is governed by the gravitational force, considered to be a constant, and the drag force due to air stream past the projectile. The earliest study on the projectile motion is traced back to the time of Galileo’s professorship at Pisa near in late 16th century [1]. The motion of a projectile under the influence of a constant gravitational force, without consideration of air resistance, is given by a parabola.

The drag force exerted on a moving object immersed in a fluid is expressed in generic terms by

F D ∝ v ^γ

with γ a positive constant whose value might be of con- troversy. For large bodies moving at relatively large speeds, the motion of the fluid molecules becomes er- ratic, the fluid flow past the body becomes turbulent, and an unordered wake is formed behind the body [2].

In this circumstance, the Reynolds number becomes of

∗

E-mail: [email protected]

the order of 10 ³ or higher and a difference in fluid pres- sure between the front and the rear of the body causes the drag force on the body [3]. Its magnitude is propor- tional to the square of the speed of the body relative to the fluid [4–6]. Many instances such as a thrown stone or a moving vehicle correspond to this category and, for this reason, some authors refer to the case of F _D ∝ v ² as

‘standard’ drag force [7].

When the velocity of a body moving in the fluid is not that large, on the other hand, the fluid stream might be viewed as laminar flow, and a thin boundary layer of fluid is formed around the body. The resistance experienced by the body is due to friction that arises as one layer of the fluid flows over another. Under such a circumstance, the drag force is linearly proportional to the velocity [8].

Our study in this paper is concerned with this class of drag force due to air through which a projectile flies with a speed in such a way that the Reynolds number be small [9].

The science of a projectile motion is of interest in

sports games that throw an object such as shotput, ham-

mer, discus, and javelin. The natural concern is to

look for the angle that would give rise to the maximum

range. To find the desired launching angle, given an ini-

tial speed, lots of endeavors have been made [10–14]. As

-870-

x

= v

τ cos θ Asymptotic x Coordinate

for the method of analysis, there have been some notable developments employing ‘fractional derivative’ [15] and a Lagrangian multiplier [16, 17], to name a few. There was also a study for the launching speed and angle with directions towards ‘affordance’ of the player [18]. All in all, however, the results are mostly provided in implicit fashion and the actual implementation issue of looking for the desired launching angle and thereby achieving the goal of maximum range is hardly found, for which this paper is intended.

The organization of this paper is as follows. After providing an analysis of the projectile motion under the influence of linear drag force due to air in section II, the launching angle for the maximum range will be looked for numerically in section III. From a numerical inves- tigation of curve-fittings for the angle vs. the involved parameter, the actual implementation of achieving the maximum range of a projectile will also be provided.

Concluding remarks will finally given in section IV. Ta- ble 1 is the list of the variables and their meanings that will be used in this paper.

II. PROJECTILE MOTION UNDER THE INFLUENCE OF LINEAR DRAG FORCE

OF AIR

We consider a projectile launched at the origin with speed v ₀ and launching angle θ. The initial conditions are then given by

r 0 = (0, 0) (1)

v 0 = v 0 (cos θ, sin θ) (2)

τ ≡ m b

that has the dimension of time. Along with this, we define the following parameters:

v T ≡ gτ = mg b y g ≡ v T τ = gτ ²

β ≡ v ₀ v T

= bv ₀ mg (x V , y V ) ≡ v 0 τ (cos θ, sin θ)

v T stands for the terminal speed in vertical motion. The parameter β denotes the ratio of the drag force of air and the projectile weight and will play a crucial role in this paper. The parameters above provide neat expressions of the equations and permit easy grasp of the dimensional analysis.

The following inequalities all represent the condition that the drag force is much less than the weight of the projectile:

β  1 bv 0  mg

v 0  v T

On the other hand, β > 1 means that the drag force of air dominates the gravitational force. We consider the range of β = [0, 5] in this paper. For β > 5, the projectile was found to advance forwards little due to the air resistance, irrespective of the specific values of β.

The equation of motion (3) is now expressed as

¨ x + x ˙

τ = 0 (4a)

¨ y + y ˙

τ = −g (4b)

Fig. 1. Plots of (a) x(t) and (b) y(t), respectively, for v 0

= 9.9 m/s and θ = 45 ^◦ . The two cases of β = 0 and β

= 4 are as labelled in the graphs. Other curves are for the values of β with increments of ∆β = 0.2.

Integration of (4a) twice gives us τ ˙ x + x = x V

x(t) = x V

 1 − exp



− t τ



(5) with the initial conditions (1) and (2) met. We see that the horizontal coordinate approaches x → x V as t → ∞.

The homogeneous equation of (4b), i.e.,

¨ y + y ˙

τ = 0 (6)

has the same form as (4a), but, in this case, the two inte- gration constants should be reserved to meet the initial conditions. The special solution for (4b) is given by

y _S (t) = −gτ t = −v _T t (7) and the complete solution of (4b) is given by the sum of (7) and the solution to (6):

y(t) = (y _g + y _V )

 1 − exp



− t τ



− y g

t

τ (8)

Fig. 2. The trajectories for various values of β with v 0

= 9.9 m/s and θ = 45 ^◦ . The two extreme cases of β = 0 and β = 4 are as labelled in the graph. For other curves, the values of β are varied with ∆β = 0.2.

Fig. 3. Range vs. x V for various values of β with v 0 = 9.9 m/s and θ = 45 ^◦ .

with the initial conditions (1) and (2) met. Fig. 1 shows plots of x(t) (Eq. (5)) and y(t) (Eq. (8)), respectively, for v 0 = 9.9 m/s and θ = 45 ^◦ . The two extreme cases of β = 0 and β = 4 are as labelled in the graph. For other curves, the values of β are varied with ∆β = 0.2.

The trajectory of the projectile is obtained by elimi- nating t in (5) and (8), the result being

y(x) = (y _g + y _V ) x x V

+ y _g ln

 1 − x

x V



(9) Figure 2 shows trajectories for various values of β. The initial speed is v 0 = 9.9 m/s and the launching angle is θ = 45 ^◦ as in Fig. 1.

The ‘range’ is determined as the root of the equation y = 0 with y(x) given by (9), i.e.,

(y _g + y _V ) x x V

+ y _g ln

 1 − x

x V



= 0 (10)

both x V and the range decrease towards zero, meaning that the projectile advances little.

The x coordinate for the maximum height (peak) is obtained by applying dy/dx = 0 to (9), the result being

x P = y V

y g + y V

x V

The maximum height can be obtained by putting this value of x back into (9):

y _P = y _V − y g ln

 1 + y _V

y g



The horizontal and vertical coordinates (5) and (8) can be expanded as follows:

x(t) = x V

"

t τ − 1

2

 t τ

 2

+ 1 6

 t τ

 3

− · · ·

#

y(t) = y V

t

τ + (y g + y V )

×

"

− 1 2

 t τ

 ² + 1

6

 t τ

 ³

− · · ·

#

Neglecting terms of O((t/τ ) ³ ), the motion is that of a constant acceleration with

a = − 1

τ ² (x V , y g + y V ) = −g ˆ y − v 0

τ

This means that the drag force might be treated as a constant force −bv 0 for t  τ or, equivalently, x(t)  x _V .

The trajectory as given by (9) may be expanded as

y(x) = y _V

 x x _V



− y _g

"

1 2

 x x _V

 ² + 1

3

 x x _V

 ³ + · · ·

#

where the first two leading terms correspond to the case of no air resistance and the remaining ones reflect the effect of the drag force of air.

Fig. 4. The launching angle θ for the maximum range vs. the parameter β.

Fig. 5. (Color online) The curve-fitting results of the exponential and power functions for θ(β) of Figure 4.

III. LAUNCHING ANGLE FOR THE MAXIMUM RANGE

By varying the launching angle θ and calculating the range from Eq. (10), the launching angle for the maxi- mum range can be evaluated numerically. Fig. 4 shows the result. β = 0 is the case of no air resistance. It is noteworthy that the result is independent of the initial speed and the mass of the projectile.

To express θ of Fig. 4 as a function of β, we try the exponential and power functions:

θ(β) = p ₀ exp(−p ₁ β) θ(β) = q ₀ β ^q

with p 0 , p 1 , q 0 , and q 1 adjustable parameters for the best

fit. Fig. 5 shows the curve-fitting results. It seems that

the above two models are not appropriate in representing

θ(β) of Fig. 4.

Fig. 6. (Color online) The curve-fitting result for θ(β) of Figure 4 by the 2nd and the 3rd order polynomials.

Table 2. The fitting coefficients for the polynomial θ(β) of Eq. (11).

Order M k

k

k

k

k

2 43.4050 -8.0508 0.7819

3 44.4557 -10.6045 2.0619 -0.1707

4 44.7993 -12.0102 3.3344 -0.5672 0.0397

As an alternative, we consider a polynomial of order M :

θ =

M

X

m=0

k m β ^m (11)

with k m adjustable parameters for the best fit. Fig. 6 shows the result of the 2nd and the 3rd order curve- fittings.

Though the 2nd order fit shows a minor deviation from the data of Fig. 4, it is almost undiscernible to distin- guish the curves of the 3rd and higher order curve-fittings from the data of Fig. 4. The fitting coefficients of Eq.

(11) are given in Table 2.

The actual implementation of obtaining the maximum range of a projectile proceeds as follows. (1)Measure the terminal speed v _T from the vertical free fall motion.

(2) Measure the launching speed v 0 . (3) Calculate the parameter β = v ₀ /v _T . (4) Get the launching angle θ from (11) with the coefficients as given in Table 2.

Figure 7 shows, as an illustrative example, the trajec- tories for various values of launching angle θ with β = 2 and v 0 = 20 m/s. The one for the maximum range is obtained for θ = 30.2 ^◦ and denoted by the thick solid

Fig. 7. The trajectories for various values of launching angle θ with β = 2 and v ₀ = 20 m/s.

line. The other curves are for θ = 30.2 ^◦ ∼ 40.2 ^◦ in steps of ∆θ = 2 ^◦ .

In the analysis until now, it has been assumed that the release and the landing points of the projectile have the same elevation. In the sports game such as javelin throwing, the height of the release point of the projectile should also be considered. In this case, the equation for the range is given by y(x) = −H, i.e.,

(y _g + y _V ) x

x _V + y _g ln

 1 − x

x _V



+ H = 0

instead of Eq. (10). Here, H is the height of the release point measured from the ground. Though an analytic expression of θ for the maximum range of this equation can be obtained in the case of no drag force of air [19], it is a formidable task to look for θ(β, H) numerically with another free parameter H added. This problem will not be pursued in this paper.

IV. CONCLUSION

In this paper, a study was performed to obtain the launching angle that gives rise to the maximum range of a projectile under the influence of linear drag force of air.

We introduced a parameter that represents the ratio of

the drag force and the projectile weight. Through a nu-

merical analysis of the trajectory, the desired launching

angle for the maximum range was obtained as a func-

tion of the parameter. The functional dependence of

the launching angle vs. the parameter was not appro-

priately expressed by either exponential or power func-

tions. Instead, it might be represented by polynomials

1996), p. 110.

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