# 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.

#### Key Points

• Objects that are projected from, and land on the same horizontal surface will have a vertically symmetrical path.

• The time it takes from an object to be projected and land is called the time of flight. This depends on the initial velocity of the projectile and the angle of projection.

• When the projectile reaches a vertical velocity of zero, this is the maximum height of the projectile and then gravity will take over and accelerate the object downward.

• The horizontal displacement of the projectile is called the range of the projectile, and depends on the initial velocity of the object.

• If an object is projected at the same initial speed, but two different angles of projection, the range of the projectile will be the same.

#### Terms

• Exhibiting symmetry; having harmonious or proportionate arrangement of parts; having corresponding parts or relations.

• The path of a body as it travels through space.

#### Figures

1. ##### Maximum Height

The maximum height is reached when the velocity is equal to zero

2. ##### Range of Trajectory

The range of a trajectory is shown in this figure.

3. ##### Projectiles at an Angle

This video gives a clear and simple explanation of how to solve a problem on Projectiles Launched at an Angle. I try to go step by step through this difficult problem to layout how to solve it in a super clear way. 2D kinematic problems take time to solve, take notes on the order of how I solved it. Best wishes. Tune into my other videos for more help. Peace.

### 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.

### Initial Velocity

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

<equation contenteditable="false">$\small{\sf{u_x = u*g*h_o*cos\theta}}\\ \small{\sf{u_y = u*sin\theta}}$

In this equation, u stands for initial velocity, g refers to gravity,  ho to initial height, and θ 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, T depends on the initial velocity and the angle of the projectile, θ:

$\small{\sf{T=\frac{2*u_y}{g}}}\\ \small{\sf{T=\frac{2*u*sin\theta}{g}}}$

### Acceleration

In projectile motion, there is no acceleration in the horizontal direction. The acceleration, a, in the vertical direction is just due to gravity, also known as free fall

ax=0

ay=-g

### Velocity

The horizontal velocity remains constant, but the vertical velocity increases linearly, because the acceleration is constant. At any time, t, the velocity is:

$v_x = u cos({\theta}) vy = u sin ({\theta}) - g*t$

Where v refers to velocity at time, t. You can also use the Pythagorean Theorem to find velocity:

$\small{\sf{v=\sqrt{v_x^2+v_y^2}}}$

### Displacement

At time, t, the displacement components are:

$\small{\sf{x=u*t*cos\theta}}\\ \small{\sf{y=u*t*sin\theta-\frac12gt^2}}$

The equation for the magnitude of the displacement is $\small{\sf{\Delta}r=\sqrt{x^2+y^2}}$.

### Parabolic Trajectory

The equation of a parabola is $y=ax+bx^2$. We can use the displacement equations in the x and y direction to obtain an equation for the parabolic form of a projectile motion:

$\small{\sf{y=tan\theta *x-\frac{g}{2*u^2*cos^2\theta}*x^2}}$

### Maximum Height

The maximum height is reached when vy=0 (Figure 1). Using this we can rearrange the velocity equation to find the time it will take for the object to reach maximum height

$\small{\sf{t_h=\frac{u*sin\theta}{g}}}$

where th stands for the time it takes to reach maximum height. From the displacement equation we can find the maximum height

$\small{\sf{h=\frac{u^2*sin^2\theta}{2*g}}}$.

### Range

The range of the motion can be found by using the following equation

$\small{\sf{R=\frac{u^2*sin2\theta}{g}}}$(Figure 2).

Figure 3

#### Key Term Glossary

acceleration
The amount by which a speed or velocity increases (and so a scalar quantity or a vector quantity).
##### Appears in these related concepts:
Acceleration
the rate at which the velocity of a body changes with time
##### Appears in these related concepts:
atom
The smallest possible amount of matter which still retains its identity as a chemical element, now known to consist of a nucleus surrounded by electrons.
##### Appears in these related concepts:
Component
A part of a vector. For example, horizontal and vertical components.
##### Appears in these related concepts:
displacement
A vector quantity that denotes distance with a directional component.
##### Appears in these related concepts:
Displacement
The length and direction of a straight line between two objects.
##### Appears in these related concepts:
equation
An assertion that two expressions are equal, expressed by writing the two expressions separated by an equal sign; from which one is to determine a particular quantity.
##### Appears in these related concepts:
force
A physical quantity that denotes ability to push, pull, twist or accelerate a body which is measured in a unit dimensioned in mass × distance/time² (ML/T²): SI: newton (N); CGS: dyne (dyn)
##### Appears in these related concepts:
Force
A force is any influence that causes an object to undergo a certain change, either concerning its movement, direction or geometrical construction.
##### Appears in these related concepts:
gravity
Resultant force on Earth's surface, of the attraction by the Earth's masses, and the centrifugal pseudo-force caused by the Earth's rotation.
##### Appears in these related concepts:
interference
An effect caused by the superposition of two systems of waves, such as a distortion on a broadcast signal due to atmospheric or other effects.
##### Appears in these related concepts:
magnitude
A number assigned to a vector indicating its length.
##### Appears in these related concepts:
motion
A change of position with respect to time.
##### Appears in these related concepts:
symmetrical
Exhibiting symmetry; having harmonious or proportionate arrangement of parts; having corresponding parts or relations.
##### Appears in these related concepts:
trajectory
The path of a body as it travels through space.
##### Appears in these related concepts:
velocity
A vector quantity that denotes the rate of change of position with respect to time, or a speed with a directional component.
##### Appears in these related concepts:
Velocity
The rate of change of displacement with respect to change in time.