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 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 is a parabola. (b) Sketch the trajectory if $\alpha=30^{\circ}$ and $v_{0}=1000 \mathrm{m} / \mathrm{s}$. (GRAPH CAN'T COPY)

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 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 is a parabola.
(b) Sketch the trajectory if $\alpha=30^{\circ}$ and $v_{0}=1000 \mathrm{m} / \mathrm{s}$.
(GRAPH CAN'T COPY)

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Key Concepts

Parabolic Trajectory

The trajectory of a projectile under gravity forms a parabola because the horizontal motion is linear while the vertical motion is quadratic in nature due to the constant gravitational force. Understanding this concept is vital in predicting and analyzing the path of objects launched into the air.

Projectile Motion

This concept refers to the motion of an object that is launched into the air and affected only by gravity, resulting in a path that combines a constant horizontal velocity with an accelerating vertical component. It is a fundamental topic in kinematics and is crucial for understanding motions such as the one described in the problem.

Parametric Equations

Parametric equations express the coordinates of the moving object as functions of a separate independent variable, typically time. This approach is essential in describing projectile motion since it provides a clear representation of how the horizontal and vertical positions evolve independently over time.

Elimination of the Parameter

Eliminating the parameter involves removing the time variable from a set of parametric equations, thereby establishing a direct relationship between the spatial coordinates. This step is key for transforming the temporal description into a spatial one, revealing the shape of the trajectory.

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