How Projectile Motion Works

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How Projectile Motion Works—Projectile motion is a type of motion in which an object, called a projectile, is thrown into the air and is then subject only to the force of gravity. This motion is commonly seen in sports such as baseball and soccer, in scientific experiments, and in military applications. In this article, we will explore the principles of projectile motion, including how it is affected by factors such as initial velocity, launch angle, and air resistance. We will also understand the equations of projectile motion that are used to calculate range, maximum altitude, and time of flight. By the end of this article, you will understand how projectile motion works, and how it can be applied to a variety of real-life situations.

Contents hide

1 Principles of Projectile Motion

2 Initial Velocity, Angle of Projection, and Air Resistance

3 Initial velocity, angle of projection, and air resistance

4 The mathematical equations used to calculate the various components of projectile motion

5 Two-Dimensional Projectile Motion

6 Formula of Initial Velocity

7 Equations of Projectile Motion

8 Formulas for projectile motion:

8.1 Time of flight:

8.2 Range

8.3 Maximum Height

9 Examples of Projectile Motion:

10 How to Do Projectile Motion Problems:

10.1 Example Problem:

11 Projectile Motion in Physics Classroom

12 Projectile Motion Lab

13 Projectile Motion Worksheets

14 Derivation of formulae of Projectile Motion

14.1 Time of flight

14.2 Maximum height

14.3 Range

14.4 Equation of trajectory

15 Projectile Motion Calculator

Principles of Projectile Motion

The principles of projectile motion are based on two independent motions: horizontal motion and vertical motion.

Horizontal motion: It is uniform and continuous, meaning the projectile moves in a straight line with constant velocity. This motion is not affected by gravity.

Vertical motion: It is affected by gravity and follows the laws of uniformly accelerated motion. It starts with an initial velocity, experiences constant acceleration $-9.8𝑚/𝑠^2$ due to gravity, reaches a maximum height and then falls back to the ground.

Angle of projection and initial velocity are the two main factors that affect the path of the projectile. Both of these determine the range, maximum height, and time of flight.

Air resistance is also an important factor affecting the motion of the projectile. It is a force that acts opposite to the motion of the projectile and can reduce the projectile’s projected range.

To understand and predict the motion of a projectile, various mathematical equations such as the equation of motion, range equation, and equation of time of flight are used. These equations take into account various factors affecting projectile motion and allow accurate prediction of the projectile’s motion.

In summary, the principles of projectile motion are based on two independent motions, horizontal and vertical, and the projectile’s motion is affected by launch angle, initial velocity, and air resistance.

Initial Velocity, Angle of Projection, and Air Resistance

The initial velocity is the velocity of the projectile the moment it is launched. It can be represented by magnitude and direction. The magnitude of the initial velocity determines the distance the projectile will travel, while the direction determines the projectile’s path. The higher the initial velocity, the farther the projectile will travel and the higher the height it will reach.

The Angle of Projection is the angle at which the projectile is launched. It affects the projectile’s path, determining the maximum height and range. At a high Angle of Projection, the projectile will reach a maximum height but travel a shorter distance, while at a lower angle, the projectile will travel a greater distance.

Air resistance is a force that acts opposite to the motion of a projectile. It is caused by the friction between the surface of the projectile and the air. Air resistance reduces the range and maximum altitude of the projectile. Depending on the size and shape of the projectile, air resistance may be significant or negligible.

In summary, initial velocity, launch angle, and air resistance are the main factors affecting the motion of a projectile. Initial velocity determines the distance the projectile travels, launch angle affects maximum altitude and range, and air resistance affects range and maximum altitude by opposing the motion of the projectile.

Initial velocity, angle of projection, and air resistance

Initial velocity is the velocity of a projectile at the instant of its projection. The initial velocity of a projectile can be represented by its magnitude and direction. The magnitude of the initial velocity determines the distance that the projectile will travel, while the direction of the initial velocity determines the trajectory of the projectile. The greater the initial velocity, the further the projectile will travel, and the higher it will reach.

Angle of projection is the angle at which the projectile is launched. The angle of projection affects the trajectory of the projectile, determining the maximum height and the range of the projectile. A projectile launched at a higher angle will reach a greater maximum height and travel a shorter distance than a projectile launched at a lower angle.

Air resistance is a force that opposes the motion of a projectile. It is caused by the friction between the air and the surface of the projectile. Air resistance acts in the opposite direction to the motion of the projectile and reduces the range and maximum height of the projectile. Depending on the size and shape of the projectile, air resistance can be significant or negligible.

To sum up, the initial velocity, angle of projection, and air resistance are the main factors that affect the motion of a projectile. The initial velocity determines the distance that the projectile will travel, the angle of projection determines the maximum height and the range of the projectile, and air resistance affects the range and maximum height of the projectile by opposing the motion of the projectile.

The mathematical equations used to calculate the various components of projectile motion

There are several mathematical equations that can be used to calculate the various components of projectile motion, such as:

Horizontal displacement (x): $x = u\cos\theta\: t$ where $u\cos\theta$ is the horizontal component of the initial velocity and t is the time of flight.

Vertical displacement (y): $y = u\sin\theta \;t - \frac{1}{2}gt^2$ where $u\sin\theta$ is the vertical component of the initial velocity, g is the acceleration due to gravity, and t is the time of flight.

Time of flight (t): $t = \frac{2u\sin\theta}{g}$ where $u\sin\theta$ is the vertical component of the initial velocity and $g$ is the acceleration due to gravity.

Maximum height (H): $H = \frac{u^2\sin^2\theta}{2g}$ where $u\sin\theta$ is the vertical component of the initial velocity and $g$ is the acceleration due to gravity.

Range (R): $R = \frac{u^2\sin2\theta}{g}$ where $u\sin\theta$ is the vertical component of the initial velocity, theta is the angle of projection, and $g$ is the acceleration due to gravity.

Velocity at any point of time (v): $v = \sqrt{v_x^2 + v_y^2}$ where $v_x$ and $v_y$ are the horizontal and vertical components of the velocity respectively.

Two-Dimensional Projectile Motion

In 2D Projectile Motion, we split the motion of the object into horizontal and vertical components. Horizontal motion occurs with constant velocity, while vertical motion is affected by gravity.

Formula of Initial Velocity

Initial velocity is divided into two components:

Horizontal velocity: $u_x = u \cos\theta$
Vertical velocity: $u_y = u \sin\theta$

Here, $u$ is the initial velocity and $\theta$ is the projection angle.

Equations of Projectile Motion

Kinematic Equations for Projectile Motion:

Horizontal distance: $x = u_x t = u \cos\theta t$
Vertical distance: $y = u_y t - \frac{1}{2} g t^2 = u\sin\theta t - \frac{1}{2}gt^2$
Vertical velocity at any time $t$ : $v_{y} = u_y - g t = v\sin\theta - gt$

Here, $g$ is the gravitational acceleration.

Formulas for projectile motion:

Time of flight:

Time of Flight is the total duration of the projectile in the air. It is the time between when the object is launched and when it comes back to the ground. The determination of time period depends only on the vertical component.

Formula of Time Period:

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

Where,

$T$ = Time Period
$u_y = u \sin\theta$ = Initial Vertical Velocity
$u$ = Initial Velocity
$\theta$ = Angle of Launch
$g$ = Gravitational Acceleration (approximately $9.8 \, m/s^2)$

Range

Range is the horizontal distance that the projectile covers while it is in the air. It is the horizontal distance from the starting point of the projectile to its return to the ground.

Formula of Range:

$R = \frac{u^2 \sin 2\theta}{g}$

Where,

$R$ = Range
$u$ = Initial Velocity
$\theta$ = Angle of Projection
$g$ = Gravitational Acceleration

This formula tells us how far the object will travel.

For maximum range the angle of projection should be $\color{\blue}45^\circ$

Maximum Height

Maximum Height is the highest point that the projectile reaches during its flight. At this point the vertical velocity of the projectile becomes zero.

Formula for maximum height:

$H = \frac{u_y^2}{2g} = \frac{(u \sin\theta)^2}{2g}$

Where,
$H$ = maximum height
$u_y = u \sin\theta$ = initial vertical velocity
$u$ = initial velocity
$\theta$ = projection angle
$g$ = gravitational acceleration

These three quantities are important aspects of projectile motion and they depend on initial velocity, projection angle and gravity.

Examples of Projectile Motion:

Ball thrown in the air
Cannonball launched
Jump in water sports

How to Do Projectile Motion Problems:

Write down the known values.
Find the horizontal and vertical components of velocity.
Use the appropriate equations.
Calculate time, distance, and velocity.

Example Problem:

A ball is thrown at an angle of $45^\circ$ with the horizontal with an initial velocity of $20\, m/s$ . Determine the following:
(a) Maximum Altitude, (b) Total Time of Flight, and (c) Range

Step 1: Write down the known values

Initial velocity, $u = 20 \, m/s$
Angle of projection, $\theta = 45^\circ$
Gravity, $g = 9.8 \, m/s^2$

Step 2: Find the horizontal and vertical components of the velocity

Horizontal velocity $( u_x )$ :
$u_x = u \cos\theta = 20 \cos45^\circ$
Since $\cos45^\circ = \frac{1}{\sqrt{2}} \approx 0.7071$ ,
$u_x = 20 \times 0.7071 = 14.14 \, m/s$
Vertical velocity $( u_y )$ :
$u_y = u \sin\theta = 20 \sin45^\circ$
Since $\sin45^\circ = \frac{1}{\sqrt{2}} \approx 0.7071$ ,
$u_y = 20 \times 0.7071 = 14.14 \, m/s$

Step 3: Use the appropriate equations

(a) Determine the maximum height ( $H$ ):

Formula for maximum height:

$H = \frac{u_y^2}{2g}$

Calculations:

$H = \frac{(14.14)^2}{2 \times 9.8} = \frac{200}{19.6} = 10.20 \, m$

(b) Determine the total time of flight ( $T$ ):

Formula for time period:

$T = \frac{2 u_y}{g}$

Calculations:

$T = \frac{2 \times 14.14}{9.8} = \frac{28.28}{9.8} = 2.88 \, s$

(c) Determine the range ( $R$ ):

Formula for range:

$R = u_x \times T$

Calculations:

$R = 14.14 \times 2.88 = 40.77 \, m$

Step 4: Calculate time, distance, and velocity

Maximum height: The maximum height to which the ball reaches is $10.20 \, m$ .
Total time of flight: The ball remains in the air for $2.88\, s$ .
Limit: The ball travels a total horizontal distance of $40.77\, m$ .

In this example, we determined the important quantities of projectile motion using the known values of initial velocity and launch angle. This procedure is valid for solving any projectile motion problem.

In summary, to solve projectile motion problems:

List all the known values given in the problem.
Mathematically divide the horizontal and vertical components of the initial velocity.
Calculate the required quantities (e.g. time, height, distance) using the kinematic equations of projectile motion.
Perform all calculations carefully and interpret the results.

Practicing this method will allow you to solve projectile motion problems easily.

To practice Projectile Motion, solve various problems and check the results.

Projectile Motion in Physics Classroom

Projectile motion is a major topic in Physics Classroom. It is studied in-depth in courses like AP Physics Projectile Motion and Physics Classroom Projectile Motion. Students evaluate their understanding by taking the Projectile Motion Test.

Projectile Motion Lab

In the Projectile Motion Lab, students experience projectile motion in real life. They present their findings by writing a Projectile Motion Lab Report.

Projectile Motion Worksheets

Projectile Motion Worksheet and Projectile Motion Worksheet with Answers PDF are available for practice. These worksheets help students solve problems.

It’s important to note that these equations are based on the assumption that air resistance is negligible and the angle of projection is small. If air resistance is significant or the angle of projection is large, these equations may not be accurate.