Projectile+Motion

= = toc
 * Projectile motion** is two-dimensional motion in which an object moves in both the x- and y-direction simultaneously with constant acceleration. A projectile is an object with an initial velocity in any given direction, and follows a path determined by the effect of gravity and/or friction. In simple physics, however, air resistance is often negated. Some examples of projectile motion are a baseball being hit by a bat, a golf ball after being struck by a club, and a bullet being shot from a gun. The projectiles in these cases are the baseball, golf ball, and bullet, respectively. Projectile motion is only possible if the acceleration of the projectile remains constant, and often times projectiles have an acceleration of 0, ignoring acceleration due to gravity.

=Process=

In projectile motion, the horizontal and vertical velocities of the object are independent of each other. This means that the first thing you want to do is to split the velocity at the given angle into components:

x-component: V x = V cos θ y-component: V y = V sin θ

From the y-component, the amount of time taken for the object to reach the ground can be found, and from that you can use the x-component and the time in order to find the distance the object had traveled.

=Examples=

**Problem #1:**
Which component you start out with is determined by the information you are given. For example, if a car is driven off a cliff 100 meters high and lands 90 meters away from the foot of the cliff, how long did it take to get to the foot of the cliff?



**Problem #2:**
What was the initial velocity of the car in the above problem? (V o )

**Problem #3:**
A projectile is thrown at a 30 degree angle with an initial velocity of 5 m/s. How long does it take for it to come back down?

**Problem #4:**
A projectile is thrown at a 30 degree angle with an initial velocity of 5 m/s. How far did the projectile travel?

**Problem #5:**
There is a man on top of a building and he throws his wallet up into the air from a height of 20 meters at a 30 degree angle because he was broke and angry. The wallet has an initial velocity of 25 m/s. How long does his wallet take to hit the ground?



**Problem #6:**
In the problem above, what was the velocity at impact?

**Problem #7:**
How far did the wallet in the above problems go?

**Problem #1:**
Using the general motion equation above, we can begin to solve the problem. Since we have more information in the y-direction, we will solve the y-parameters first. Notice the **X**'s were replaced with **Y**'s. This is done to clarify the solving process by splitting the Horizontal(x) and Vertical(y) components apart. Substituting in all the numbers, we get Solving for t, we get
 * ~  ||~ Horizontal ||~ Vertical ||
 * ~ Displacement || 90.0m || 100.0m ||
 * ~ Intitial Velocity(Vo) || ? || 0m/s ||
 * ~ Acceleration || 0 || 9.8m/s/s ||
 * ~ Time || ? || ? ||



**Problem #2:**
The time is the same for both horizontal and vertical because the projectile takes the same amount of time to reach **X=0** and **Y=0**
 * ~  ||~ Horizontal ||~ Vertical ||
 * ~ Displacement || 90.0m || 100.0m ||
 * ~ Initial Velocity(Vo) || ? || 0m/s ||
 * ~ Acceleration || 0 || 9.8 ||
 * ~ Time || 4.52 s || 4.52 s ||

The last thing we need to solve for is the initial velocity in the horizontal direction. The initial velocity of the projectile in the vertical direction is 0, because the car rolled off a cliff, it wasnt dropped or thrown, which indicates a negative or positive vertical velocity.

Now back to using the horizontal form of the general motion equation, We substitute all the values we know, we get, Solving for initial velocity we get,

**Problem #3:**
A projectile is thrown at a 30 degree angle with an initial velocity of 5 m/s. How long does it take for it to come back down? We would split the velocity into to two vectors, V x and V y, and we are find V y.

Using trigonometry, V y = 5sin30 = 2.5 m/s. The V yi = 2.5 m/s, meaning that since the projectile will be half the time and then falling with the same magnitude, the V yf =-2.5m/s. ΔV y = V yf  - V yi  = -2.5 - 2.5  = -5 m/s The change in velocity is equal to the acceleration times the change in time, so we set it this way:  -5 m/s= a y (ΔT)  -5 m/s = -9.8ΔT

 ΔT = 0.510 s

**Problem #4: **
A projectile is thrown at a 30 degree angle with an initial velocity of 5 m/s. How far did the projectile travel? This is a continuation of the previous problem, and first we will find the horizontal component of the vector.

So now, using trig again, V x = 5cos30 = 4.33 m/s. We found how long the projectile was in the air in the previous problem, T = 0.510 s. We can find displacement with the formula for the change in position. X = VΔT <span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; font-size: small;">X = (4.33)(0.510) X = 2.2083 m

**Problem #5:**
There is a man on top of a building and he throws his wallet up into the air from a height of 20 meters at a 30 degree angle because he was broke and angry. The wallet has an initial velocity of 25 m/s. How long does his wallet take to hit the ground?

First we find the initial velocity components of the wallet. V ix = 25cos30 = 21.65 m/s V iy = 25sin30 = 25 m/s We find the y-displacement with y i = 0, y = -20 m, V iy = 25 m/s. <span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; font-size: small;">Δy = V iy <span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; font-size: small;">t- 1/2gt 2 <span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; font-size: small;">-20 = 25t - 4.9t 2 <span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; font-size: small;"> t = 5.81s

<span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; font-size: small;">**Problem #6**:
<span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; font-size: small;">In the problem above, what was the velocity at impact?

<span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; font-size: small;">V y <span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; font-size: small;"> = V iy <span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; font-size: small;"> - gt <span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; font-size: small;"> = 25 - 9.8(5.81) <span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; font-size: small;"> = -31.938 m/s <span style="color: #222222; font-family: arial,sans-serif;">Now that we have that, we need to find the total velocity, so <span style="color: #222222; font-family: arial,sans-serif;">V = (V x 2 <span style="color: #222222; font-family: arial,sans-serif;"> + V y 2 <span style="color: #222222; font-family: arial,sans-serif;">) 1/2 <span style="color: #222222; font-family: arial,sans-serif;"> = ((21.65) 2 <span style="color: #222222; font-family: arial,sans-serif;">+(25) 2 <span style="color: #222222; font-family: arial,sans-serif;">) 1/2 <span style="color: #222222; font-family: arial,sans-serif;"> = 33.071 m/s

**<span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; font-size: small;">Problem #7: **
How far did the wallet in the above problem go?

We just need to plug in information that we already know into the horizontal displacement equation. <span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; font-size: small;">Δx = V ix <span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; font-size: small;">t <span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; font-size: small;"> = 21.65(5.81) <span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; font-size: small;"> = 125.79 m

=References=

[] @http://www.khanacademy.org/science/physics/v/deriving-max-projectile-displacement-given-time @http://www.khanacademy.org/science/physics/v/total-final-velocity-for-projectile