# Motion in Two Dimensions

## Projectile Motion


### Section Key Terms


### Properties of Projectile Motion

Projectile motion is the motion of an object thrown (projected) into the air when, after the initial force that launches the object, air resistance is negligible and the only other force that object experiences is the force of gravity. The object is called a projectile, and its path is called its trajectory. Air resistance is a frictional force that slows its motion and can significantly alter the trajectory of the motion. Due to the difficulty in calculation, only situations in which the deviation from projectile motion is negligible and air resistance can be ignored are considered in introductory physics. That approximation is often quite accurate.
The most important concept in projectile motion is that when air resistance is ignored, horizontal and vertical motions are , meaning that they don’t influence one another.  compares a cannonball in free fall (in blue) to a cannonball launched horizontally in projectile motion (in red). You can see that the cannonball in free fall falls at the same rate as the cannonball in projectile motion. Keep in mind that if the cannon launched the ball with any vertical component to the velocity, the vertical displacements would not line up perfectly.
Since vertical and horizontal motions are independent, we can analyze them separately, along perpendicular axes. To do this, we separate projectile motion into the two components of its motion, one along the horizontal axis and the other along the vertical.
We’ll call the horizontal axis the x-axis and the vertical axis the y-axis. For notation, d is the total displacement, and x and y are its components along the horizontal and vertical axes. The magnitudes of these vectors are x and y, as illustrated in .
As usual, we use velocity, acceleration, and displacement to describe motion. We must also find the components of these variables along the x- and y-axes. The components of acceleration are then very simple a = –g = –9.80 m/s2. Note that this definition defines the upwards direction as positive. Because gravity is vertical, a = 0. Both accelerations are constant, so we can use the kinematic equations. For review, the kinematic equations from a previous chapter are summarized in .
Where x is position, x0 is initial position, v is velocity, vavg is average velocity, t is time and a is acceleration.


### Solve Problems Involving Projectile Motion

The following steps are used to analyze projectile motion:
1. Separate the motion into horizontal and vertical components along the x- and y-axes. These axes are perpendicular, so  
          and  
         are used. The magnitudes of the displacement 
         along x- and y-axes are called 
         and 
         The magnitudes of the components of the velocity 
         are 
         and 
        , where 
         is the magnitude of the velocity and 
         is its direction. Initial values are denoted with a subscript 0.
2. 
        Treat the motion as two independent one-dimensional motions, one horizontal and the other vertical. The kinematic equations for horizontal and vertical motion take the following forms
        
        Vertical motion (assuming positive is up 
           )
        
3. Solve for the unknowns in the two separate motions (one horizontal and one vertical). Note that the only common variable between the motions is time 
         . The problem solving procedures here are the same as for one-dimensional kinematics.
4. Recombine the two motions to find the total displacement 
       and velocity 
       . We can use the analytical method of vector addition, which uses 
       and 
       to find the magnitude and direction of the total displacement and velocity.
     
         is the direction of the displacement 
        , and 
         is the direction of the velocity 
         . (See 
      
The expression we found for  
       while solving part (a) of the previous problem works for any projectile motion problem where air resistance is negligible. Call the maximum height  
      ; then,
This equation defines the maximum height of a projectile. The maximum height depends only on the vertical component of the initial velocity.

### Practice Problems

The fact that vertical and horizontal motions are independent of each other lets us predict the range of a projectile. The range is the horizontal distance R traveled by a projectile on level ground, as illustrated in . Throughout history, people have been interested in finding the range of projectiles for practical purposes, such as aiming cannons.
How does the initial velocity of a projectile affect its range? Obviously, the greater the initial speed 
      , the greater the range, as shown in the figure above. The initial angle 
       also has a dramatic effect on the range. When air resistance is negligible, the range 
       of a projectile on level ground is
where 
         is the initial speed and 
         is the initial angle relative to the horizontal. It is important to note that the range doesn’t apply to problems where the initial and final y position are different, or to cases where the object is launched perfectly horizontally.

### Section Summary

1. Projectile motion is the motion of an object through the air that is subject only to the acceleration of gravity.
2. Projectile motion in the horizontal and vertical directions are independent of one another. 
3. The maximum height of an projectile is the highest altitude, or maximum displacement in the vertical position reached in the path of a projectile.
4. The range is the maximum horizontal distance traveled by a projectile.
5. To solve projectile problems: choose a coordinate system; analyze the motion in the vertical and horizontal direction separately; then, recombine the horizontal and vertical components using vector addition equations.

### Key Equations


### Check Your Understanding


### Concept Items


### Problems


### Critical Thinking


### Test Prep Multiple Choice


### Test Prep Short Answer


### Test Prep Extended Response
