# Deriving Projectile Motion Equations by Applying Kinematics Equations of Motion to Moving Bodies

## Physics, Mechanics, Kinematics and Ballistics

Physics is an area of science which deals with how matter and waves behave in the Universe. A branch of physics called mechanics deals with forces, matter, energy, work done and motion. A further sub-branch known as kinematics deals with motion and ballistics is specifically concerned with the motion of projectiles launched into the air, water or space. Solving ballistic problems involves using Newton's equations of motion.

See my article, Force, Weight, Newtons, Velocity, Mass and Friction - Basic Principles of Mechanics, which introduces the basic concepts of mechanics

In these examples, for the sake of simplicity, the effects of air friction known as *drag* have been excluded.

## What are the Equations of Motion? (SUVAT Equations)

Consider a body of mass ** m**, acted on by a force

**for time**

*F**. This produces an acceleration which we will designate with the letter*

**t***. The body has an initial velocity*

**a***, and after time*

**u***, it reaches a velocity*

**t***. It also travels a distance*

**v***.*

**s**So we have 5 parameters associated with the body in motion: u, v, a, s and t

The equations of motion allow us to work out any of these parameters once we know three other parameters. So the three most useful formulae are:

*v = u + at*

*s= ut + ½at ^{2}*

*v ^{2} = u^{2} + 2as*

*Remember, Newton's second law of motion tells us that F = ma so the acceleration of a body depends on the force. It only accelerates and increases in speed as long as a force is applied (or decelerates and decreases in speed if the force opposes motion). Once the force is removed, the velocity stays constant unless another force acts on it (Newton's first law of motion). In our examples, that other force is gravity which causes velocity to increase or decrease.*

## Solving Projectile Motion Problems — Calculating Time of Flight, Distance Traveled and Altitude

High school and college exam questions in ballistics usually involve calculating time of flight, distance traveled and altitude attained.

There are 4 scenarios normally presented in these types of problems, and it is necessary to calculate parameters mentioned above:

- Object dropped from a known altitude
- Object thrown upward
- Object thrown horizontally from a height above the ground
- Object launched from the ground at an angle

These problems are solved by considering the initial or final conditions and this enables us to work out a formula for velocity, distance traveled, time of flight and altitude. To decide which of Newton's three equations to use, check which parameters you know and use the equation with one unknown, i.e. the parameter you want to work out.

In example 3 and 4, breaking the motion down into its horizontal and vertical components allows us to find the required solutions.

## The Trajectory of Ballistic Bodies is a Parabola

Unlike guided missiles, which follow a path which is variable and controlled by pure electronics or more sophisticated computer control systems, a ballistic body such as a shell, cannon ball, particle or stone thrown into the air follows a parabolic trajectory after it is launched. The launching device (gun, hand, sports equipment etc.) gives the body an acceleration and it leaves the device with an initial velocity. The examples below ignore the effects of air drag which reduce the range and altitude attained by the body.

For lots more information on parabolas, see my tutorial:

How to Understand the Equation of a Parabola, Directrix and Focus

## 1. Free Falling Object Dropped From a Known Height

*v = u + at*

*s= ut + ½at ^{2}*

*v ^{2} = u^{2} + 2as*

In this case the falling body starts off at rest and reaches a final velocity v. The acceleration in all these problems is a = g (the acceleration due to gravity). Remember though that the sign of g is important as we will see later.

### Calculating final velocity

u = 0 (the body is initially at rest)

a = g (g is positive because it is in the direction of motion and accelerating the body)

s = h (the height the object is dropped from)

The equation *v = u + at* can't be used because t is unknown, so use the equation *v ^{2} = u^{2} + 2as*

So:

v^{2} = u^{2} + 2as

* = *0^{2} + 2gh = 2gh

Taking the square root of both sides

v = √(2gh) This is the final velocity

**Calculating instantaneous distance fallen**

s = ut + ½at^{2}

= 0t + ½gt^{2}

So s = ½gt^{2}

**Calculating time taken to fall distance h**

s = h = ut + ½at^{2}

= 0t + ½gt^{2}

So h = ½gt^{2}

Which gives

t^{2} = 2h/g

Taking square roots of both sides

t = √(2h/g)

## 2. Object Projected Vertically Upwards

*v = u + at*

*s= ut + ½at ^{2}*

*v ^{2} = u^{2} + 2as*

In this scenario, the body is vertically projected upwards at 90 degrees to the ground with an initial velocity u. The final velocity v is 0 at the point where the object reaches maximum altitude and becomes stationary before falling back to Earth. The acceleration in this case is a = -g as gravity slows down the body during its upwards motion.

Let t_{1} and t_{2} be the time of flights upwards and downwards respectively

**Calculating time of flight upwards**

v = u + at

So

0 = u + (-g)t

Giving

u = gt

So

t_{1} = u/g

### Calculating distance traveled upwards

v^{2} = u^{2} + 2as

So

0^{2} = u^{2} + 2(-g)s

So

u^{2} = 2gs

Giving

s = h = u^{2}/(2g)

### Calculating time of flight downwards

We calculated previously that the time taken for an object to fall a distance h is:

t = √(2h/g)

But we worked out above that h =u^{2}/(2g) is the distance traveled upwards

Substituting:

t_{2} = √(2h/g) = √(2(u^{2}/(2g))/g) = √(2u^{2}/2g^{2}) = u/g

This is also u/g. You can calculate it knowing the altitude attained as worked out below and knowing that the initial velocity is zero. Hint: use example 1 above!

**Total time of flight**

total time of flight is t_{1} + t_{2} = u/g + u/g = 2u/g

## 3. Object Projected Horizontally From a Height

*v = u + at*

*s= ut + ½at ^{2}*

*v ^{2} = u^{2} + 2as*

A body is horizontally projected from a height h with an initial velocity of u relative to the ground. The key to solving this type of problem is knowing that the vertical component of motion is the same as what happens in example 1 above, when the body is dropped from a height. So as the projectile is moving forwards, it is also moving downwards, accelerated by gravity

### Time of flight

t = √(2h/g) as calculated in example 1

**Distance traveled horizontally**

There is no horizontal acceleration, just a vertical acceleration g due to gravity

So distance traveled = velocity x time = ut = u√(2h/g)

So

s = u√(2h/g)

## 4. Object Projected at an Angle to the Ground

*v = u + at*

*s= ut + ½at ^{2}*

*v ^{2} = u^{2} + 2as*

In this example, a projectile is thrown at an angle Θ to the ground with an initial velocity u. This problem is the most complex, but using basic trigonometry, we can resolve the velocity vector into vertical and horizontal components. Time of flight and vertical distance traveled to the apex of the trajectory can then be calculated using the method in example 2 (object thrown upwards). Once we have the time of flight, this allows us to calculate the horizontal distance traveled during this period.

See diagram below

Let u_{h} be the horizontal component of initial velocity

Let u_{v} be the vertical component of initial velocity

So

Cos Θ = u_{h }/ u

Giving u_{h} = uCos Θ

Similarly

Sin Θ = u_{v} / u

Giving u_{v} = uSin Θ

**Time of flight to apex of trajectory**

From example 2, the time of flight is t = u/g. However since the vertical component of velocity is u_{v}

t = u_{v}/g = uSin Θ/g

**Altitude attained**

Again from example 2, the vertical distance traveled is s = u^{2}/(2g). However since u_{v} = uSin Θ is the vertical velocity:

s = u_{v}^{2}/(2g) = (uSin Θ)^{2}/(2g))

**Horizontal distance traveled**

Since the time of flight is uSin Θ/g to the apex of the trajectory and uSin Θ/g during the period when the projectile is falling back to the ground (see downward time of flight example 2)

Total time of flight is:

2uSin Θ/g

Now during this period, the projectile is moving horizontally at a velocity u_{h} = uCos Θ

So horizontal distance traveled = horizontal velocity x total time of flight

= uCos Θ x (2uSin Θ)/g

= (2u^{2}Sin ΘCos Θ)/g

The double angle formula can be used to simplify

I.e. Sin 2A = 2SinACosA

So (2u^{2}Sin ΘCos Θ)/g = (u^{2}Sin 2Θ)/g

Horizontal distance to apex of trajectory is half this or:

(u^{2}Sin 2Θ)/2g

## A Vector Can be Resolved Into Two Components

## What is the Optimum Angle to Launch a Projectile?

The optimum angle to launch a projectile is the angle which gives maximum horizontal range.

Using basic differential calculus, we can differentiate the function for horizontal range wrt θ and set it to zero allowing us to find the peak of the curve (of the graph of range versus launch angle, __not__ the peak of the actual trajectory). Then find the angle which satisfies the equation.

So horizontal range = 2u^{2}Sin θCos θ/g

Product rule:

If a'(x) is the derivative of a(x) w.r.t. x

and b'(x) is the derivative of b(x) w.r.t. x

then derivative of a(x)b(x) w.r.t x = a(x)b'(x) + b(x)a'(x)

Derivative of Sin θ is Cos θ

Derivative of Cos θ is -Sin θ

Using the product rule and isolating the constant 2u^{2}/g :

d/dΘ(2u^{2}Sin θ Cos θ/g)

= 2u^{2 }/g(d/dθ(Sin θ Cos θ))

= 2u^{2 }/g [(Sin θ)(-Sin θ) + (Cos θ)( Cos θ)]

Setting this to zero

2u^{2 }/g [(Sin θ)(-Sin θ) + (Cos θ)( Cos θ)] = 0

Divide each side by 2u^{2 }/g and rearranging gives:

(Sin θ)^{2} = (Cos θ)^{2}

So Sin θ = Cos θ

And the angle which satisfies this is θ = 45^{ο}

## Orbital Velocity Formula: Satellites and Spacecraft

What happens if an objected is projected really fast from the Earth? As the object's velocity increases, it falls further and further from the point where it was launched. Eventually the distance it travels horizontally is the same distance that the Earth's curvature causes the ground to fall away vertically. The object is said to be in *orbit.* The velocity that this happens at is approximately 25,000 km/h in low Earth orbit.

If a body is much smaller than the object it is orbiting, the velocity is approximately:

v ≈ √(GM / r)

Where M is the mass of the larger body (in this case Earth's mass)

r is the distance from the centre of the Earth

G is the gravitational constant = 6.67430 × 10 ^{−11} m^{3}⋅kg^{−1}⋅s^{−2}

If we exceed the orbital velocity, an object will escape a planet gravity and travel outwards from the planet. This is how the Apollo 11 crew were able to escape Earth's gravity. By timing the burn of rockets that provided propulsion and getting the velocities just right at the right moment, the astronauts were then able to insert the spacecraft into lunar orbit. Later in the mission as the LM was deployed, it used rockets to slow its velocity so that it dropped out of orbit, eventually culminating in the 1969 lunar landing.

## A Short History Lesson....

ENIAC (Electronic Numerical Integrator And Computer) was one of the first general purpose computers designed and built during WW2 and completed in 1946. It was funded by the U.S. Army and the incentive for its design was to enable the calculation of ballistic tables for artillery shells, taking into account the effects of drag, wind and other factors influencing projectiles in flight.

ENIAC, unlike the computers of today was a colossal machine, weighing 30 tons, consuming 150 kilowatt of power and taking up 1800 square feet of floor space. At the time it was proclaimed in the media as "a human brain". Before the days of transistors, integrated circuits and micropressors, *vacuum tubes* (also known as "valves"), were used in electronics and performed the same function as a transistor. i.e they could be used as a switch or amplifier. Vacuum tubes were devices which looked like small light bulbs with internal filaments which had to be heated up with an electrical current. Each valve used a few of watts of power, and since ENIAC had over 17,000 tubes, this resulted in huge power consumption. Also tubes burnt out regularly and had to be replaced. 2 tubes were required to store 1 bit of information using a circuit element called a *"flip-flop"* so you can appreciate that the memory capacity of ENIAC was nowhere near what we have in computers today.

ENIAC had to be programmed by setting switches and plugging in cables and this could take weeks.

## Engineering Mathematics by K.A. Stroud

A groundbreaking and comprehensive reference with over 500,000 copies sold since it first debuted in 1970, the new seventh edition of Engineering Mathematics has been thoroughly revised and expanded. An interactive Personal Tutor CD-ROM is included with every book. Providing a broad mathematical survey, this innovative volume covers a full range of topics from the very basic to the advanced.

I highly recommend this book as it's like having a personal tutor who guides you through the solution of problems as it progresses through the text and explains everything clearly.

## Questions & Answers

An object is projected from velocity u=30 m/s making an angle of 60°. How do I find height, range and flight time of object if g=10?

u = 30 m/s

Θ = 60°

g = 10 m/s²

height = (uSin Θ)2/(2g))

range = (u2Sin 2Θ)/g

time of flight = uSin Θ/g

Plug the numbers above into the equations to get the results.

Helpful 98If I am to find how high an object rises, should I use the 2nd or 3rd equation of motion?

Use v² = u² + 2as

You know the initial velocity u, and also velocity is zero when the object reaches max height just before it starts to fall again. The acceleration a is -g. The minus sign is because it acts in the opposite direction to the initial velocity U, which is positive in the upwards direction.

Helpful 45An object is fired from the ground at 100 meters per seconds at an angle of 30 degrees with the horizontal how high is the object at this point?

If you mean the maximum altitude attained, use the formula (uSin Θ)²/(2g)) to work out the answer.

u is the initial velocity = 100 m/s

g is the acceleration due to gravity a 9.81 m/s/s

Θ = 30 degrees

Helpful 17

**© 2014 Eugene Brennan**

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