Basic Equations and Parabolic Path

Projectile motion is a form of motion where an object moves in parabolic path; the path that the object follows is called its trajectory.

Projectile Motion

Projectile motion is a form of motion where an object moves in a bilaterally symmetrical, parabolic path. The path that the object follows is called its trajectory. Projectile motion only occurs when there is one force applied at the beginning on the trajectory, after which the only interference is from gravity. In a previous atom we discussed what the various components of an object in projectile motion are. In this atom we will discuss the basic equations that go along with them in the special case in which the projectile initial positions are null (i.e. [latex]\text{x}_0 = 0[/latex] and [latex]\text{y}_0 = 0[/latex] ).

Initial Velocity

The initial velocity can be expressed as x components and y components:

[latex]\text{u}_\text{x} = \text{u} \cdot \cos\theta \\ \text{u}_\text{y} = \text{u} \cdot \sin\theta[/latex]

In this equation, [latex]\text{u}[/latex] stands for initial velocity magnitude and [latex]\small{\theta}[/latex] refers to projectile angle.

Time of Flight

The time of flight of a projectile motion is the time from when the object is projected to the time it reaches the surface. As we discussed previously, [latex]\text{T}[/latex] depends on the initial velocity magnitude and the angle of the projectile:

[latex]\displaystyle {\text{T}=\frac{2 \cdot \text{u}_\text{y}}{\text{g}}\\ \text{T}=\frac{2 \cdot \text{u} \cdot \sin\theta}{\text{g}}}[/latex]

Acceleration

In projectile motion, there is no acceleration in the horizontal direction. The acceleration, [latex]\text{a}[/latex], in the vertical direction is just due to gravity, also known as free fall:

[latex]\displaystyle {\text{a}_\text{x} = 0 \\ \text{a}_\text{y} = -\text{g}}[/latex]

Velocity

The horizontal velocity remains constant, but the vertical velocity varies linearly, because the acceleration is constant. At any time, [latex]\text{t}[/latex], the velocity is:

[latex]\displaystyle {\text{u}_\text{x} = \text{u} \cdot \cos{\theta} \\ \text{u}_\text{y} = \text{u} \cdot \sin {\theta} - \text{g} \cdot \text{t}}[/latex]

You can also use the Pythagorean Theorem to find velocity:

[latex]\text{u}=\sqrt{\text{u}_\text{x}^2+\text{u}_\text{y}^2}[/latex]

Displacement

At time, t, the displacement components are:

[latex]\displaystyle {\text{x}=\text{u} \cdot \text{t} \cdot \cos\theta\\ \text{y}=\text{u} \cdot \text{t} \cdot \sin\theta-\frac12\text{gt}^2}[/latex]

The equation for the magnitude of the displacement is [latex]\Delta \text{r}=\sqrt{\text{x}^2+\text{y}^2}[/latex].

Parabolic Trajectory

We can use the displacement equations in the x and y direction to obtain an equation for the parabolic form of a projectile motion:

[latex]\displaystyle \text{y}=\tan\theta \cdot \text{x}-\frac{\text{g}}{2 \cdot \text{u}^2 \cdot \cos^2\theta} \cdot \text{x}^2[/latex]

Maximum Height

The maximum height is reached when [latex]\text{v}_\text{y}=0[/latex]. Using this we can rearrange the velocity equation to find the time it will take for the object to reach maximum height

[latex]\displaystyle \text{t}_\text{h}=\frac{\text{u} \cdot \sin\theta}{\text{g}}[/latex]

where [latex]\text{t}_\text{h}[/latex] stands for the time it takes to reach maximum height. From the displacement equation we can find the maximum height

[latex]\displaystyle \text{h}=\frac{\text{u}^2 \cdot \sin^2\theta}{2\cdot \text{g}}[/latex]

Range

The range of the motion is fixed by the condition [latex]\small{\sf{\text{y} = 0}}[/latex]. Using this we can rearrange the parabolic motion equation to find the range of the motion:

[latex]\displaystyle \text{R}=\frac{\text{u}^2 \cdot \sin2\theta}{\text{g}}[/latex].

Solving Problems

In projectile motion, an object moves in parabolic path; the path the object follows is called its trajectory.

We have previously discussed projectile motion and its key components and basic equations. Using that information, we can solve many problems involving projectile motion. Before we do this, let’s review some of the key factors that will go into this problem-solving.

What is Projectile Motion?

Projectile motion is when an object moves in a bilaterally symmetrical, parabolic path. The path that the object follows is called its trajectory. Projectile motion only occurs when there is one force applied at the beginning, after which the only influence on the trajectory is that of gravity.

What are the Key Components of Projectile Motion?

The key components that we need to remember in order to solve projectile motion problems are:

• Initial launch angle, [latex]\theta[/latex]
• Initial velocity, [latex]\text{u}[/latex]
• Time of flight, [latex]\text{T}[/latex]
• Acceleration, [latex]\text{a}[/latex]
• Horizontal velocity, [latex]\text{v}_\text{x}[/latex]
• Vertical velocity, [latex]\text{v}_\text{y}[/latex]
• Displacement, [latex]\text{d}[/latex]
• Maximum height, [latex]\text{H}[/latex]
• Range, [latex]\text{R}[/latex]

Now, let’s look at two examples of problems involving projectile motion.

Zero Launch Angle

An object launched horizontally at a height [latex]\text{H}[/latex] travels a range [latex]\text{v}_0 \sqrt{\frac{2\text{H}}{\text{g}}}[/latex] during a time of flight [latex]\text{T} = \sqrt{\frac{2\text{H}}{\text{g}}}[/latex].

Projectile motion is a form of motion where an object moves in a parabolic path. The path followed by the object is called its trajectory. Projectile motion occurs when a force is applied at the beginning of the trajectory for the launch (after this the projectile is subject only to the gravity).

One of the key components of the projectile motion, and the trajectory it follows, is the initial launch angle. The angle at which the object is launched dictates the range, height, and time of flight the object will experience while in projectile motion. shows different paths for the same object being launched at the same initial velocity and different launch angles. As illustrated by the figure, the larger the initial launch angle and maximum height, the longer the flight time of the object.

We have previously discussed the effects of different launch angles on range, height, and time of flight. However, what happens if there is no angle, and the object is just launched horizontally? It makes sense that the object should be launched at a certain height ([latex]\text{H}[/latex]), otherwise it wouldn’t travel very far before hitting the ground. Let’s examine how an object launched horizontally at a height [latex]\text{H}[/latex] travels. In our case is when [latex]\alpha[/latex] is 0.

Duration of Flight

There is no vertical component in the initial velocity ([latex]\text{v}_0[/latex]) because the object is launched horizontally. Since the object travels distance [latex]\text{H}[/latex] in the vertical direction before it hits the ground, we can use the kinematic equation for the vertical motion:

[latex](\text{y}-\text{y}_0) = -\text{H} = 0\cdot \text{T} - \frac{1}{2} \text{g} \text{T}^2[/latex]

Here, [latex]\text{T}[/latex] is the duration of the flight before the object its the ground. Therefore:

[latex]\displaystyle \text{T} = \sqrt{\frac{2\text{H}}{\text{g}}}[/latex]

Range

In the horizontal direction, the object travels at a constant speed [latex]\text{v}_0[/latex] during the flight. Therefore, the range [latex]\text{R}[/latex] (in the horizontal direction) is given as:

[latex]\displaystyle \text{R}= \text{v}_0 \cdot \text{T} = \text{v}_0 \sqrt{\frac{2\text{H}}{\text{g}}}[/latex]

General Launch Angle

The initial launch angle (0-90 degrees) of an object in projectile motion dictates the range, height, and time of flight of that object.

Projectile motion is a form of motion where an object moves in a bilaterally symmetrical, parabolic path. The path that the object follows is called its trajectory. Projectile motion only occurs when there is one force applied at the beginning of the trajectory, after which the only interference is from gravity.

One of the key components of projectile motion and the trajectory that it follows is the initial launch angle. This angle can be anywhere from 0 to 90 degrees. The angle at which the object is launched dictates the range, height, and time of flight it will experience while in projectile motion. shows different paths for the same object launched at the same initial velocity at different launch angles. As you can see from the figure, the larger the initial launch angle, the closer the object comes to maximum height and the longer the flight time. The largest range will be experienced at a launch angle up to 45 degrees.

The range, maximum height, and time of flight can be found if you know the initial launch angle and velocity, using the following equations:

[latex]\small{\sf{\text{R}=\frac{\text{v}_\text{i}^2\sin2\theta_\text{i}}{\text{g}}}}\\ \small{\sf{\text{h}=\frac{\text{v}_\text{i}^2\sin^2\theta_\text{i}}{2\text{g}}}}\\ \small{\sf{\text{T}=\frac{2\text{v}_\text{i}sin\theta}{\text{g}}}}[/latex]

Where R – Range, h – maximum height, T – time of flight, v_i – initial velocity, θ_i – initial launch angle, g – gravity.

Now that we understand how the launch angle plays a major role in many other components of the trajectory of an object in projectile motion, we can apply that knowledge to making an object land where we want it. If there is a certain distance, d, that you want your object to go and you know the initial velocity at which it will be launched, the initial launch angle required to get it that distance is called the angle of reach. It can be found using the following equation:

[latex]\small{\sf{\theta=\frac12sin^{-1}(\frac{\text{gd}}{\text{v}^2})}}[/latex]

Key Points: Range, Symmetry, Maximum Height

Projectile motion is a form of motion where an object moves in parabolic path. The path that the object follows is called its trajectory.

What is Projectile Motion ?

Projectile motion is a form of motion where an object moves in a bilaterally symmetrical, parabolic path. The path that the object follows is called its trajectory. Projectile motion only occurs when there is one force applied at the beginning on the trajectory, after which the only interference is from gravity. In this atom we are going to discuss what the various components of an object in projectile motion are, we will discuss the basic equations that go along with them in another atom, “Basic Equations and Parabolic Path”

Key Components of Projectile Motion:

Time of Flight, T:

The time of flight of a projectile motion is exactly what it sounds like. It is the time from when the object is projected to the time it reaches the surface. The time of flight depends on the initial velocity of the object and the angle of the projection, [latex]\theta[/latex]. When the point of projection and point of return are on the same horizontal plane, the net vertical displacement of the object is zero.

Symmetry:

All projectile motion happens in a bilaterally symmetrical path, as long as the point of projection and return occur along the same horizontal surface. Bilateral symmetry means that the motion is symmetrical in the vertical plane. If you were to draw a straight vertical line from the maximum height of the trajectory, it would mirror itself along this line.

Maximum Height, H:

The maximum height of a object in a projectile trajectory occurs when the vertical component of velocity, [latex]\text{v}_\text{y}[/latex], equals zero. As the projectile moves upwards it goes against gravity, and therefore the velocity begins to decelerate. Eventually the vertical velocity will reach zero, and the projectile is accelerated downward under gravity immediately. Once the projectile reaches its maximum height, it begins to accelerate downward. This is also the point where you would draw a vertical line of symmetry.

Range of the Projectile, R:

The range of the projectile is the displacement in the horizontal direction. There is no acceleration in this direction since gravity only acts vertically. shows the line of range. Like time of flight and maximum height, the range of the projectile is a function of initial speed.