Problem 41 - Precalculus Mathematics for Calculus

Question

If a projectile is fired with an initial speed of \(v_{0}\) fts at an angle \(\alpha\) above the horizontal, then its position after \(t\) seconds is given by the parametric equations $$x=\left(v_{0} \cos \alpha\right) t \quad y=\left(v_{0} \sin \alpha\right) t-16 t^{2}$$ (where \(x\) and \(y\) are measured in feet). Show that the path of the projectile is a parabola by eliminating the parameter \(t\).

Step-by-Step Solution

Answer

The path is a parabola because the y-equation is a quadratic in x.

Step 1: Write the Parametric Equations

The motion's position after time \( t \) is given by the parametric equations:\[x = (v_0 \cos \alpha) t \quad \text{and} \quad y = (v_0 \sin \alpha) t - 16t^2.\]

Step 2: Solve for t in terms of x

We start by manipulating the equation for \( x \) to solve for \( t \):\[ x = (v_0 \cos \alpha) t \]\[ t = \frac{x}{v_0 \cos \alpha}. \]

Step 3: Substitute t in y equation

Take the expression for \( t \) derived in Step 2 and substitute it into the \( y \) equation:\[y = \left(v_0 \sin \alpha\right) \left(\frac{x}{v_0 \cos \alpha}\right) - 16 \left(\frac{x}{v_0 \cos \alpha}\right)^2.\]

Step 4: Simplify the y equation

Simplify the expression by performing the multiplication and reduction:\[ y = x \tan \alpha - 16 \frac{x^2}{(v_0 \cos \alpha)^2}. \]Notice that this is now a quadratic equation in terms of \( x \), confirming that it represents a parabola.

Key Concepts

Projectile Motion

Projectile motion involves the motion of an object that is launched into the air, subject only to the forces of gravity and air resistance. In this exercise, a projectile is fired with an initial speed and at an angle above the horizontal. The motion of the projectile can be described by two parametric equations, one for the horizontal position and one for the vertical position. This tells you how far the projectile travels in the x direction and how high it goes in the y direction over time.

• The horizontal motion is captured by the equation: \[x = (v_0 \cos \alpha)t,\] where \(v_0\) is the initial speed and \(\alpha\) is the angle of projection.
• This means that the horizontal velocity remains constant since there is no acceleration in the x-direction.
• Gravity affects the vertical motion, modifying the y-equation to: \[y = (v_0 \sin \alpha)t - 16t^2.\]
Here, gravity acts downwards, hence the term \(16t^2\) which makes the motion curvilinear instead of straight.

Parabola

A parabola is a specific type of curve that is symmetric and has a characteristic U-shape. In the context of projectile motion, the path of the projectile can be shown to be a parabola by eliminating the time parameter and examining the relationship between x and y.

• The trajectory of a projectile under constant acceleration due to gravity is always a parabola.
• This is because the vertical motion is influenced by a squared time term, seen in the \( -16t^2 \) part of the y equation.
• When you eliminate the time parameter, the resultant equation will contain a quadratic term in relation to the horizontal distance (x), which defines a parabola's equation like this: \[y = x \tan \alpha - 16 \frac{x^2}{(v_0 \cos \alpha)^2}.\]
This quadratic equation confirms that the projectile's course is parabolic.

Eliminating the Parameter

Eliminating the parameter refers to removing the time factor from the parametric equations to produce a direct relationship between the x (horizontal) and y (vertical) positions. This is crucial because it allows us to see the projectile's path as a single equation rather than two separate ones.

• First, solve \(t\) in terms of \(x\) using the horizontal position equation: \[t = \frac{x}{v_0 \cos \alpha}.\]
• Substitute this expression for \(t\) in the vertical position equation, which transforms it into a direct relationship between \(x\) and \(y\).
• The resulting equation, \[y = x \tan \alpha - 16 \frac{x^2}{(v_0 \cos \alpha)^2},\] demonstrates the motion of the projectile without explicitly involving time.

This expression highlights the parabolic nature of the trajectory, achieved by eliminating the time parameter.