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2013-09-27T15:51:04Z

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==Projectile Motion Basics==

''Helena Dedic''

• A particle is in projectile motion when the horizontal component of its velocity is different from zero and when the gravitational force is the only force acting on it. 
◦ We will always choose a coordinate system with y-axis pointing upward and x-axis pointing along the horizontal.
• Its acceleration is then equal to g and points downward. Therefore: 

<math>a_x = 0</math> 
<math>a_y = - 10 m/s^2 </math> 

◦ The x-component of its velocity is constant. The <math>v_x - t</math> graph is a horizontal straight line. Therefore: 

<math>v_x(t) = v_{0x}</math> 
<math>\Delta x(t) = v_{0x}t</math> 

◦ The y-component of its velocity is decreasing at a rate of 10 m/s every second. It is a linear function of time and its <math>v_y - t</math> graph is a straight line with a negative slope of<math> - 10 m/s^2</math>. Therefore: 

<math>v_y(t) = v_{0y} - 10t</math> 
<math>\Delta y(t) = v_{0y}t - 5t^2</math> 

• The velocity at any time t is a vector: 

<math>\vec v(t) = v_{0x} \hat i + (v_{0y} - 10t) \hat j</math> 

• The path of an object in projectile motion is a parabola. At the vertex of the parabola, the y-component of the velocity is zero and thus the velocity is: 

<math>\vec v(t) = v_{0x} \hat i</math> 

• When the vertical displacement is zero then the horizontal displacement is called range R. The range depends only on the initial speed of the projectile and on the direction of the velocity that is the angle <math>\theta</math> above horizontal. 

<math>R = \frac{v_i^2 Sin 2\theta}{g} </math> 
 
*[[#Projectile Motion Basics|TOP]] 

==Derivation of the Equation for Horizontal Range==

We begin with the notion that the velocity has a magnitude v0 and the direction  above horizontal. Then the component of the initial velocity are: 

<math> v_{0x} = v_0 Cos\theta</math> 
<math> v_{0y} = v_0 Sin\theta</math> 

We also note that <math>\Delta x(t) = R</math> and <math>\Delta y(t) = 0</math>. We will substitute these values into the equations of kinematics for projectile motion: 

<math>R = v_0 (Cos\theta) t</math> 

<math>v_y(t) = v_0 Sin\theta - 10 t</math> 

<math>0 = v_0 (Sin\theta)t - 5 t^2</math> 

We isolate t from the third equation above:

<math>0 = v_0 (Sin\theta)</math><strike>t</strike>- <math>5 t</math> 
<math> t = \frac{v_0 Sin\theta}{5}</math> 

and substitute for t in the first equation for the range: 

<math>R = v_0 (Cos\theta)\frac{v_0 Sin\theta}{5} = \frac{v_0^2 Sin\theta Cos\theta}{5}</math> 

We note the trigonometric identity <math>sin2\theta = 2 sin\theta cos\theta</math>. We can substitute for <math>sin\theta cos\theta</math> in the equation for the range and obtain: 

<math>R = \frac{v_0^2 Sin2\theta /2}{5} = \frac{v_0^2 Sin2\theta}{10}</math> 

So, in general: 

<math>R = \frac{v_0^2 Sin2\theta}{g}</math> 
 

*[[#Projectile Motion Basics|TOP]] 

==Related Videoclips==

 

<ol>
<li>[http://www.youtube.com/v/N0H-rv9XFHk Range of Projectile]</li>
See how the horizontal range of a projectile depends on the initial angle of incidence. The initial velocity is the same in each case.

:::::<youtube>N0H-rv9XFHk</youtube>

<li>[http://www.youtube.com/watch?v=z24_ihikEqQ&NR=1 Parabolic motion versus vertical free fall]</li>

Compare parabolic motion of a projectile to vertical free fall motion.

:::::<youtube>z24_ihikEqQ</youtube>

<li> [http://www.youtube.com/watch?v=ZBfy-MNgtoY Projectile Motion: velocity vector]</li>

Velocity vector (along with its x and y components)as it changes in Projetile motion.

:::::<youtube>ZBfy-MNgtoY</youtube>

<li> [http://www.youtube.com/watch?v=cxvsHNRXLjw Monkey and a Gun]</li>

The monkey falls downward at the same rate as the bullet. The acceleration due to gravity is the same for both, even though the bullet is fired with a high initial velocity.

 

<li> [http://www.youtube.com/watch?v=-uUsUaPJUc0&NR=1&feature=endscreen Lecture on Projectile Motion]</li>

:::::<youtube>-uUsUaPJUc0</youtube>

 
<li> [http://www.youtube.com/watch?v=HRFbruIXVaY&feature=related Solving Projectile Motion Problems]</li>

:::::<youtube>HRFbruIXVaY</youtube>

 
</ol>
 

*[[#Projectile Motion Basics|TOP]] 

==Simulations==

[http://phet.colorado.edu/sims/projectile-motion/projectile-motion_en.html Phet Projectile Motion Simulation] 

[http://www.pbs.org/opb/circus/classroom/circus-physics/activity-guide-projectile-motion/ Circus Physics: Projectile motion]

==Exercises==

*[[Exercises on Projectile Motion]] 

<center> [[#Projectile Motion Basics|TOP]] </center>

Projectile Motion - Revision history

Luoq: /* Simulations */