# How to Solve Projectile Motion Problems: Applying Newton's Equations of Motion to Ballistics

*Eugene is a qualified control/instrumentation engineer Bsc (Eng) and has worked as a developer of electronics & software for SCADA systems.*

## 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 and is specifically concerned with the motion of projectiles launched into the air, water or space. Solving ballistic problems involves using the kinematics equations of motion, also known as the SUVAT equations or Newton's equations of motion.

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

*See also my article, Force, Weight, Newtons, Velocity, Mass and Friction – Basic Principles of Mechanics, which introduces the basic concepts.*

## 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 five 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 applied F and its mass m. The body 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 of the body 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 four basic scenarios normally presented in these types of problems, and it is necessary to calculate the 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 examples three and four, 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 football, shell, cannonball, stone or any other object projected 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

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

*v = u *+* at*

*s= ut *+ ½*at*^{2}

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

*as*

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}*+ 2

*as*

So:

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

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

Taking the square root of both sides

v = √(2gh) This is the final velocity

**Calculating Instantaneous Distance Fallen**

*s* = ut + ½*at*^{2}

= 0t + ½*gt*^{2}

So s = ½*gt*^{2}

**Calculating Time Taken to Fall Distance h**

*s* = *h* = *ut* + ½*at*^{2}

= 0*t* + ½*gt*^{2}

So *h* = ½*gt*^{2}

Which gives

*t*^{2} = 2*h*/*g*

Taking square roots of both sides

*t* = √(2*h*/*g*)

## Example 2. Object Projected Vertically Upwards

*v = u *+* at*

*s= ut *+ ½*at*^{2}

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

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*be the time of flights upwards and downwards respectively

_{2}**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} + 2*as*

So

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

So

*u*^{2} = 2*gs*

Giving

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

### Calculating Time of Flight Downwards

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

*t* = √(2*h*/g)

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

Substituting:

*t _{2}* = √(2

*h*/

*g*) = √(2(

*u*

^{2}/(2

*g*))/

*g*) = √(2

*u*

^{2}/2

*g*

^{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*= 2

*u*/

*g*

## Example 3. Object Projected Horizontally From a Height

*v = u *+* at*

*s= ut *+ ½*at*^{2}

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

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* = √(2*h*/*g*) as calculated in example one

**Distance Traveled Horizontally**

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

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

So

*s* = *u*√(2*h*/*g*)

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

*v = u *+* at*

*s= ut *+ ½*at*^{2}

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

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.

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}* =

*u*cos

*θ*

Similarly

sin *θ* = *u*_{v} / *u*

Giving *u _{v}* =

*u*sin

*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 =

*u*sin

*θ*/

*g*

**Altitude Attained**

Again from example 2, the vertical distance traveled is *s* = *u*^{2}/(2g). However since *u _{v}* =

*u*sin

*θ*is the vertical velocity:

*s* = *u _{v}*

^{2}/(2

*g*) = (

*u*sin

*θ*)

^{2}/(2

*g*))

**Horizontal Distance Traveled**

Since the time of flight is *u*sin *θ*/*g* to the apex of the trajectory and *u*sin *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:

2*u *sin *θ*/*g*

Now during this period, the projectile is moving horizontally at a velocity *u _{h}* =

*u*cos

*θ*

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

= *u *cos *θ* x (2*u *sin *θ*)/*g*

= (2*u*^{2}sin *θ c*os *θ*)/*g*

The double angle formula can be used to simplify

I.e. sin 2*A* = 2sin*A*cos*A*

So (2*u*^{2}sin *θc*os *θ*)/*g* = (*u*^{2}sin 2*θ*)/*g*

Horizontal distance to apex of trajectory is half this or:

(*u*^{2}sin 2*θ*)/2*g*

## Recommended Books

### Mathematics

*Engineering Mathematics* by KA Stroud is an excellent math textbook for both engineering students and anyone with an interest in the subject. The material has been written for part one of BSc. Engineering Degrees and Higher National Diploma courses.

A wide range of topics are covered, including some we used in this article (vectors and calculus), matrices, complex numbers, calculus applications, differential equations, series, probability theory, and statistics. The text is written in the style of a personal tutor, guiding the reader through the content, posing questions, and encouraging them to provide the answer.

I highly recommend this book. It makes learning mathematics fun!

### Mechanics

*Applied Mechanics* by John Hanah and MJ Hillier is a standard text book for students taking Diploma and Technician courses in engineering. It covers concepts used in this article (vectors, velocity and acceleration) as well as as other topics such as motion in a circle, periodic motion, statics and frameworks, impulse and momentum, stress and strain, bending of beams and fluid dynamics. Worked examples are included in each chapter in addition to set problems with answers provided.

## 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 = (*u*^{2}sin 2*θ*)/*g*

Rearranging and separating out the constant gives us

*u*^{2}/*g* (sin 2*θ*)

We can use the function of a function rule to differentiate sin 2*θ*

So if we have a function *f*(*g*), and *g* is a function of *x*, i.e. *g*(*x*)

*f '*(*g*) is the derivative of *f *(*g*) wrt *g* and *g'*(*x*) is the derivative of *g* wrt *x*

Then *f '*(*x*) = *f '*(*g*)* g'*(*x*)

So to find the derivative of sin 2*θ*, we differentiate the "outer" function giving cos 2*θ* and multiply by the derivative of 2*θ* giving 2, so

*d/dθ*(sin 2θ) = 2cos 2*θ*

Returning to the equation for range, we need to differentiate it and set it to zero to find the max range.

Using the multiplication by a constant rule

*d/dθ* (*u*^{2}/*g* (sin 2*θ*)) = *u*^{2}/*g* *d/dθ* (sin 2*θ*)

= *u*^{2}/*g* (2cos 2*θ*)

Setting this to zero

*u*^{2}/*g* (2cos 2*θ*) = 0

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

cos 2*θ* = 0

And the angle which satisfies this is 2*θ* = 90°

So *θ* = 90/2 = 45°

## 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 ≈ √(*G*M / 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's 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 kilowatts of power and taking up 1,800 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 microprocessors, *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 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.

## References

Stroud, K.A., (1970) *Engineering Mathematics *(3rd ed., 1987). Macmillan Education Ltd., London, England.

Hannah, J. and Hillerr, M. J., (1971) *Applied Mechanics* (First metric ed. 1971). Pitman Books Ltd., London, England.

Lissauer, J. J., & Pater, D. I. (2019). In *Fundamental planetary science: Physics, Chemistry and Habitability* (pp. 29–31). essay, Cambridge University Press.

*This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.*

## Questions & Answers

**Question:** 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?

**Answer:** u = 30 m/s

Θ = 60°

g = 10 m/s²

height = (uSin Θ)²/(2g))

range = (u²Sin (2Θ) )/g

time of flight to apex of trajectory = uSin Θ/g

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

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

**Answer:** 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 upward direction.

v² = u² + 2as giving 0² = u² - 2gs

Rearranging 2gs = u²

So s = √(u²/2g)

**Question:** An 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?

**Answer:** 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

**© 2014 Eugene Brennan**

## Comments

**Eugene Brennan (author)** from Ireland on December 09, 2019:

The equations above assume that a projectile is travelling in a vacuum and ignore the effects of air resistance or drag. So horizontal velocity is constant and unchanging and vertical velocity in an upwards direction only reduces due to gravity and increases ad finitum when an object is falling.The drag equation gives the force which opposes motion and it's necessary to solve a differential equation to determine actual velocity, which reduces to zero because of drag. The drag equation includes a factor for fluid density and drag is greater for higher density. As regards buoyancy, this will add a constant upwards force in the force equation.

**Kam** on December 08, 2019:

Is it possible to link these equations to the density if the fluid is not air, for example, water and linked to the law of buoyancy?

**BHARATH** on November 10, 2019:

Hi Sir,

There are only 11 branches of physics or more if it's more can you post the branch of physics

**poojitha** on November 06, 2019:

can u give detailed examples ?

**dinkineh abinet** on July 30, 2019:

.I think I have got s/thing it is really helpful thank you

**Nweke Solomon Chimaobi** on June 27, 2019:

iam an 0'level holder, thanks to you engineer Eugene Brennan for refreshing my brain , am very glad for visiting this webite. congratulations for your endless effort.

**ojhadv** on May 20, 2014:

its quite a refreshing article to read back class 7 physics again,

I had always used to throw a cricket ball to calculate its speed depending on distance after i learnt these equations, made physics interesting