Chapter 8, Problem 6P

Shooting into the Wind Suppose that a projectile is fired into a headwind that pushes it back so as to reduce its horizontal speed by a constant amount w. Find parametric equations for the path of the projectile.

To determine

The parametric equation for the path of the projectile if the horizontal velocity is decreased by a constant factor $\omega$ due to the headwind.

Answer to Problem 6P

The parametric equations for the path of projectile affected by headwind are $x=\left(v_o\cos \theta -\omega \right)t$ and $y=\left(v_o\sin \theta \right)t-\frac{1}{2}g{t}^2$.

Explanation of Solution

Given:

The projectile is fired into a headwind that reduces its horizontal velocity by a constant factor $\omega$.

Calculation:

The path of the projectile has two velocity components, a horizontal and a vertical component.

Suppose that the projectile is fired into the headwind with an initial velocity $v_0$ and at an angle $\theta$ from the ground.

Since only the horizontal velocity is decreased by a constant factor $\omega$ due to the effect of headwind, the net horizontal velocity of the projectile is $v_x=v_o\cos \theta -\omega$.

The vertical velocity remains unchanged and is equal to, $v_y=v_0\sin \theta$.

The horizontal distance covered by the projectile is, $\text{distance}=\text{velocity}\times \text{time}$.

Substitute $v_o\cos \theta -\omega$ for velocity, x for distance and t for time in the above formula.

$x=\left(v_o\cos \theta -\omega \right)t$

The vertical velocity is not affected by the headwind, but it is affected by the acceleration due to gravity.

Use the second equation of motion to find the vertical distance covered by the projectile as follows.

The second equation of motion is $s=ut+\frac{1}{2}a{t}^2$, where u is the initial velocity of the body, $a$ is acceleration of the body, $s$ is the displacement of the body and $g$ is acceleration due to gravity.

Substitute $v_o\sin \theta$ for $u$, $y$ for $s$ and $-g$ for $a$ in the above equation.

$\begin{array}{l}y=v_o\sin \theta \times t-\frac{1}{2}g{t}^2\\ y=\left(v_o\sin \theta \right)t-\frac{1}{2}g{t}^2\end{array}$

Hence, the parametric equations for the path of projectile affected by the headwind are $x=\left(v_o\cos \theta -\omega \right)t$ and $y=\left(v_o\sin \theta \right)t-\frac{1}{2}g{t}^2$.