1. Parabolic Projectile on a Level Field

Projectile motion near the Earth’s surface, assuming no air resistance, follows a parabolic trajectory. This means that the only force acting on the projectile once it’s launched is gravity, which points vertically downward. We break the motion into horizontal ($x$) and vertical ($y$) components, each governed by different aspects of kinematics.

1.1 Equations of Motion

Consider an initial launch speed $v_0$ at an angle $\theta$ above the horizontal. Gravity acts downward with an acceleration of $g \approx 9.8\,\mathrm{m\,s^{-2}}$. If we ignore air resistance, the motion equations in each direction become:

Explanation of the terms:

• $v_0\cos(\theta)$ is the constant horizontal velocity (no horizontal acceleration).
• $v_0\sin(\theta)$ is the initial vertical velocity component.
• $-\tfrac{1}{2}g\,t^2$ accounts for the displacement due to uniform downward acceleration by gravity.

These equations assume the projectile starts at $(x=0,\, y=0)$ when $t=0$. If the launch or landing positions differ in height, additional terms or different boundary conditions may apply, but the basic method of splitting the motion into components remains the same.

1.2 Key Results

By analyzing the $x(t)$ and $y(t)$ equations, we can derive simpler expressions for flight duration, maximum height, and horizontal range (assuming the landing height is the same as the launch height).

1. Time of Flight ($T$)
   $$T = \frac{2\,v_0\,\sin(\theta)}{g}.$$

   This result comes from setting $y(T) = 0$ (the projectile returns to ground level) and solving for $T$. Notice that $v_0 \sin(\theta)$ is the initial upward velocity. Dividing by $g$ gives the time to go up, and multiplying by 2 accounts for both the ascent and descent.

2. Maximum Height ($H$)
   $$H = \frac{v_0^2\,\sin^2(\theta)}{2\,g}.$$

   This expression is found by considering that at the top of the flight, the vertical velocity is zero. The vertical component of the initial kinetic energy converts into potential energy at the peak. Alternatively, you can set the derivative $\frac{dy}{dt}=0$ at the peak to find the time at maximum height, then plug that time back into $y(t)$.

3. Range on Level Ground ($R$)
   $$R = \frac{v_0^2\,\sin(2\theta)}{g}.$$

   The range formula shows that the horizontal distance depends on $\sin(2\theta)$, meaning the optimal launch angle (ignoring air resistance) is $\theta=45^\circ$. This formula assumes the projectile lands at the same vertical height from which it was launched.

2. Resultant Force Vector Calculation

When multiple forces act on an object, each force is a vector that has both magnitude and direction. The net effect on the object is determined by the vector sum of these forces— often referred to as the resultant force or the net force. Vector addition ensures we account correctly for both the directions and magnitudes.

2.1 Vector Summation

Let the individual forces be $\vec{F}_1, \vec{F}_2, \dots, \vec{F}_n$, each of which can be decomposed into $x$- and $y$-components. If force $i$ has magnitude $|\vec{F}_i|$ and is oriented at an angle $\theta_i$ from the positive $x$-axis, then:

The net force components in $x$ and $y$ are found by simply summing:

This means we add up all the $x$-direction contributions to get $F_{\mathrm{net},x}$, and all the $y$-direction contributions to get $F_{\mathrm{net},y}$. The resultant vector is:

This technique works in exactly the same way as standard 2D vector addition. The only difference is that here each vector specifically represents a physical force.

2.2 Resultant Magnitude and Direction

Once $F_{\mathrm{net},x}$ and $F_{\mathrm{net},y}$ are known, you can reconstruct the net force’s magnitude and direction. Think of it like using the Pythagorean theorem in the $xy$-plane:

and the angle (measured from the positive $x$-axis) can be found via the inverse tangent:

However, be aware that $\tan^{-1}$ (or “arctan”) alone might not indicate the correct quadrant if $F_{\mathrm{net},x}$ is negative. You should check the signs of $F_{\mathrm{net},x}$ and $F_{\mathrm{net},y}$ to decide whether to add $180^\circ$ (or $\pi$ radians) to the result.

Combining these techniques—understanding projectile motion and resultant forces—allows physicists and engineers to analyze a wide variety of real-world scenarios, from the arc of a thrown ball to the equilibrium or acceleration of an object under multiple applied forces.