ArtsAutosBooksBusinessEducationEntertainmentFamilyFashionFoodGamesGenderHealthHolidaysHomeHubPagesPersonal FinancePetsPoliticsReligionSportsTechnologyTravel

Projectile Motion Equations for Intro Physics

Updated on October 6, 2016

A particle moves in a vertical plane with an initial velocity vo, but its acceleration is always the free-fall acceleration g= 9.8m/s2, which is directed downward. Such a particle is called a projectile, and it's motion is called projectile motion. Many sports deal with this sort of motion. Let's take a look at how to solve these problems. Firstly note, that we assume air resistance is negligible aka is not considered into the problem.

The projectile is launched with an initial velocity vo:


The components vox and voy can be found if we know the angle Θo between vo and the positive x-direction:

vox=vocosΘo and voy=vosinΘo

During it's two-dimensional motion, the projectile position vector, r, and velocity, v, changes throughout but it's acceleration vector, a, is constant and always directed vertically downward. The projectile has no horizontal acceleration. And neither the horizontal nor vertical motion affect the other- they are independent of each other.

We break up a projectile problem into two simpler problems: horizontal (acceleration= 0m/s2) and vertical (constant downward acceleration= 9.8m/s2)

Now that we have that information we can start analyzing projectile motion.

Horizontal Motion

Again: There is no horizontal acceleration, so a=0m/s2. We can start with equation 1.

(1) Δx=v0t+ (1/2)at2

Since we know a=0m/s2, we can see that equation 1 changes to 1a.

(1a) Δx=v0t

(1a) (x-x0)= v0t

For the horizontal component, we also know that vox=vocosΘo. Subbing that into 1a, we get another equation we can now plug numbers into, 1b.

(1b) Δx= (v0cosΘ0)t

(1b) (x-x0)= (v0cosΘ0)t

Vertical Motion

For the vertical motion we know that the acceleration is free-fall acceleration directed downward. This means that a=-g throughout the projectile motion, for the vertical component. (a= -9.8m/s2). We can also use the same equation we used for the horizontal component of motion, but in terms of y, since are dealing with the vertical motion. See equation 2 and 2a.

(2) Δy=v0t+ (1/2)at2

(2a) (y-y0)=v0t+ (1/2)at2

For the vertical component, we know that voy=vosinΘo, therefore we get the equation in 2b and 2c.

(2b) Δy=(v0sinΘ0)t+ (1/2)(-g)t2

(2c) (y-y0)=(v0sinΘ0)t - (1/2)gt2

The equations

(3) vf=vo+ at


(4) vf2=vo2+ 2aΔx

will also be useful equations when dealing with the vertical motion of a projectile problem and can be transformed. When plugging in what we know for the acceleration and velocity for vertical motion, as discussed earlier, we can get the equations in 3a and 4a.

(3a) vy= v0sinΘ0 - gt


(4a) vy2= (v0sinΘ0)2- 2gΔy

The velocity is directed upward initially and the magnitude of the velocity steadily decreases until it reaches 0. Once it reaches 0m/s, it has reached the maximum height.

The Trajectory

If we solve for t in equation 1 and substitute that into equation 2, with a little rearranging we can get equation 5, which is the path of the projectile, aka it's trajectory.

(5) y= (tanΘ0)x- (gx2/[2(v0cosΘ0)2]

The Horizontal Range

The horizontal range is the distance the projectile traveled horizontally. In equation 1, instead of having (x-xo) we will have R for range, as shown in equation 6. And since we are looking at the horizontal range (distance along x), in equation 2, we know that (y-yo=0), as shown in equation 7.

(6) R= (v0cosΘ0)t


(7) 0= (v0sinΘ0)- (1/2)gt2

The next thing we can do is eliminate t from bother equations 6 and 7, producing equation 8.

(8) R= ([2v02]/ g)sinΘ0 cosΘ0

Next we have to use the identity0= 2sinΘ0cosΘ0 and from that we obtain equation 9, representing the range.

(9) R= ([v02]/g) sin 2Θ0

*This equation is for the range of a projectile when the initial launch height is equivalent to the final launch height.


Submit a Comment

No comments yet.


This website uses cookies

As a user in the EEA, your approval is needed on a few things. To provide a better website experience, uses cookies (and other similar technologies) and may collect, process, and share personal data. Please choose which areas of our service you consent to our doing so.

For more information on managing or withdrawing consents and how we handle data, visit our Privacy Policy at:

Show Details
HubPages Device IDThis is used to identify particular browsers or devices when the access the service, and is used for security reasons.
LoginThis is necessary to sign in to the HubPages Service.
Google RecaptchaThis is used to prevent bots and spam. (Privacy Policy)
AkismetThis is used to detect comment spam. (Privacy Policy)
HubPages Google AnalyticsThis is used to provide data on traffic to our website, all personally identifyable data is anonymized. (Privacy Policy)
HubPages Traffic PixelThis is used to collect data on traffic to articles and other pages on our site. Unless you are signed in to a HubPages account, all personally identifiable information is anonymized.
Amazon Web ServicesThis is a cloud services platform that we used to host our service. (Privacy Policy)
CloudflareThis is a cloud CDN service that we use to efficiently deliver files required for our service to operate such as javascript, cascading style sheets, images, and videos. (Privacy Policy)
Google Hosted LibrariesJavascript software libraries such as jQuery are loaded at endpoints on the or domains, for performance and efficiency reasons. (Privacy Policy)
Google Custom SearchThis is feature allows you to search the site. (Privacy Policy)
Google MapsSome articles have Google Maps embedded in them. (Privacy Policy)
Google ChartsThis is used to display charts and graphs on articles and the author center. (Privacy Policy)
Google AdSense Host APIThis service allows you to sign up for or associate a Google AdSense account with HubPages, so that you can earn money from ads on your articles. No data is shared unless you engage with this feature. (Privacy Policy)
Google YouTubeSome articles have YouTube videos embedded in them. (Privacy Policy)
VimeoSome articles have Vimeo videos embedded in them. (Privacy Policy)
PaypalThis is used for a registered author who enrolls in the HubPages Earnings program and requests to be paid via PayPal. No data is shared with Paypal unless you engage with this feature. (Privacy Policy)
Facebook LoginYou can use this to streamline signing up for, or signing in to your Hubpages account. No data is shared with Facebook unless you engage with this feature. (Privacy Policy)
MavenThis supports the Maven widget and search functionality. (Privacy Policy)
Google AdSenseThis is an ad network. (Privacy Policy)
Google DoubleClickGoogle provides ad serving technology and runs an ad network. (Privacy Policy)
Index ExchangeThis is an ad network. (Privacy Policy)
SovrnThis is an ad network. (Privacy Policy)
Facebook AdsThis is an ad network. (Privacy Policy)
Amazon Unified Ad MarketplaceThis is an ad network. (Privacy Policy)
AppNexusThis is an ad network. (Privacy Policy)
OpenxThis is an ad network. (Privacy Policy)
Rubicon ProjectThis is an ad network. (Privacy Policy)
TripleLiftThis is an ad network. (Privacy Policy)
Say MediaWe partner with Say Media to deliver ad campaigns on our sites. (Privacy Policy)
Remarketing PixelsWe may use remarketing pixels from advertising networks such as Google AdWords, Bing Ads, and Facebook in order to advertise the HubPages Service to people that have visited our sites.
Conversion Tracking PixelsWe may use conversion tracking pixels from advertising networks such as Google AdWords, Bing Ads, and Facebook in order to identify when an advertisement has successfully resulted in the desired action, such as signing up for the HubPages Service or publishing an article on the HubPages Service.
Author Google AnalyticsThis is used to provide traffic data and reports to the authors of articles on the HubPages Service. (Privacy Policy)
ComscoreComScore is a media measurement and analytics company providing marketing data and analytics to enterprises, media and advertising agencies, and publishers. Non-consent will result in ComScore only processing obfuscated personal data. (Privacy Policy)
Amazon Tracking PixelSome articles display amazon products as part of the Amazon Affiliate program, this pixel provides traffic statistics for those products (Privacy Policy)
ClickscoThis is a data management platform studying reader behavior (Privacy Policy)