We will show later in the text that if a projectile is fired...

We will show later in the text that if a projectile is fired from ground level with an initial speed of $v_{0}$ meters per second at an angle $\alpha$ with the horizontal, and if air resistance is neglected, then its position after $t$ seconds, relative to the coordinate system in the accompanying figure on the next page is $$ x=\left(v_{0} \cos \alpha\right) t, \quad y=\left(v_{0} \sin \alpha\right) t-\frac{1}{2} g t^{2} $$ where $g \approx 9.8 \mathrm{~m} / \mathrm{s}^{2}$ (a) By eliminating the parameter, show that the trajectory lies on the graph of a quadratic polynomial. (b) Use a graphing utility to sketch the trajectory if $\alpha=30^{\circ}$ and $v_{0}=1000 \mathrm{~m} / \mathrm{s}$. (c) Using the trajectory in part (b), how high does the shell rise? $\quad$ (cant.). (d) Using the trajectory in part (b), how far does the shell travel horizontally?

We will show later in the text that if a projectile is fired from ground level with an initial speed of $v_{0}$ meters per second at an angle $\alpha$ with the horizontal, and if air resistance is neglected, then its position after $t$ seconds, relative to the coordinate system in the accompanying figure on the next page is
$$
x=\left(v_{0} \cos \alpha\right) t, \quad y=\left(v_{0} \sin \alpha\right) t-\frac{1}{2} g t^{2}
$$
where $g \approx 9.8 \mathrm{~m} / \mathrm{s}^{2}$
(a) By eliminating the parameter, show that the trajectory lies on the graph of a quadratic polynomial.
(b) Use a graphing utility to sketch the trajectory if $\alpha=30^{\circ}$ and $v_{0}=1000 \mathrm{~m} / \mathrm{s}$.
(c) Using the trajectory in part (b), how high does the shell rise? $\quad$ (cant.).
(d) Using the trajectory in part (b), how far does the shell travel horizontally?

Key Concepts

Projectile Motion

Projectile motion describes the movement of an object thrown into the air under the influence of gravity, ignoring air resistance. The motion is analyzed by separating it into horizontal and vertical components, with the horizontal motion at constant speed and the vertical motion subject to constant acceleration due to gravity.

Parameter Elimination

Parameter elimination involves removing the independent parameter (often time) from a system of equations to obtain a direct relationship between the dependent variables. In the context of projectile motion, it is used to derive the trajectory equation by expressing vertical displacement directly in terms of horizontal displacement.

Quadratic Function

The trajectory of a projectile, upon elimination of time, is represented by a quadratic function. This quadratic equation models a parabolic path, which is characteristic of projectile motion when air resistance is neglected.

Optimization (Maximum Height)

Optimization in the context of projectile motion refers to finding the maximum value of the vertical displacement function. This is typically done by determining the vertex of the quadratic function, which indicates the highest point reached by the projectile.

Horizontal Range Calculation

The horizontal range is the total distance traveled by the projectile along the horizontal axis before landing. It is derived by finding the time of flight and using the constant horizontal velocity to compute the distance, making use of the quadratic trajectory equation.

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