Deriving the Equation for an Oblique Projectile Given Horizontal Range and Maximum Height

Understanding the motion of an oblique projectile involves deriving mathematical equations based on its initial velocity, launch angle, and the gravitational acceleration. In this article, we will derive the equation for the range R and the maximum height H of an oblique projectile, and explore the relationships between these parameters.

Equations of Motion for an Oblique Projectile

The motion of an oblique projectile can be described by its horizontal and vertical components. Let the initial velocity be u, and let it make an angle A with the horizontal. The equations for the projectile's motion in the horizontal (x) and vertical (y) directions are as follows:

HORIZONTAL DIRECTION (x-axis): x u cos{A} t VERTICAL DIRECTION (y-axis): y u sin{A} t - frac{1}{2} g t^2

Deriving the Horizontal Range

The R, the horizontal range of the projectile, can be found by setting the vertical component of the projectile's motion to zero at the moment of impact:

y 0 when x R

Substituting the expression for t from the horizontal motion equation into the vertical motion equation, we get:

y u sin{A} left(frac{x}{u cos{A}}right) - frac{1}{2} g left(frac{x}{u cos{A}}right)^2

Substituting t frac{x}{u cos{A}} into the vertical equation yields:

y u sin{A} frac{x}{u cos{A}} - frac{1}{2} g left(frac{x}{u cos{A}}right)^2

Simplifying the above expression, we get:

y x tan{A} - frac{1}{2} g frac{x^2}{u^2 cos^2{A}}

Maximum Height of the Projectile

The maximum height H is reached when the vertical velocity is zero. This occurs when:

frac{dy}{dx} 0

Substituting the simplified equation of vertical motion and differentiating with respect to x, we find the value of x at the maximum height:

0 frac{dy}{dx} tan{A} - frac{2x tan{A}}{R}

Solving for x, we get:

x frac{R}{2}

Relating Horizontal Range and Maximum Height

Substituting x frac{R}{2} back into the equation for y 0, we find the maximum height H:

H left(frac{R}{2}right) tan{A} - frac{1}{4} frac{R^2}{u^2 cos^2{A}}

Solving for tan{A}, we get:

tan{A} frac{4H}{R}

Substituting tan{A} frac{4H}{R} back into the equation for the range:

R u^2 sin{2A} frac{g}{2u^2 cos^2{A}} frac{gR}{2} frac{sin{2A}}{cos^2{A}} frac{Rg}{sin{2A}}

Where sin{2A} 2 sin{A} cos{A}. Therefore, the relationship between R and H can be expressed as:

H frac{R tan{A}}{4}

Conclusion

The equations derived for the range R and the maximum height H of an oblique projectile provide a deeper understanding of its motion. By manipulating these equations, we can determine the parameters of the projectile based on its range and maximum height. This knowledge is invaluable in various fields, including physics, engineering, and sports science.