

Lesson 1: Vectors  Fundamentals and OperationsRelative Velocity and Riverboat Problems
Independence of Perp. Components of
Motion Lesson 2: Projectile MotionCharacteristics of a Projectile's Trajectory Describing Projectiles with Numbers: Horizontal and Vertical Velocity Horiz. Launched Projectiles  Problem Solving
NonHoriz. Launched Projectiles 
Problem Solving Lesson 3 : Forces in Two Dimensions

Lesson 2: Projectile MotionDescribing Projectiles With Numbers(Horizontal and Vertical Displacement) The previous diagrams, tables, and discussion pertain to how the horizontal and vertical components of the velocity vector change with time during the course of projectile's trajectory. Now we will investigate the manner in which the horizontal and vertical components of a projectile's displacement vary with time. As has already been discussed, the vertical displacement (denoted by the symbol y in the discussion below) of a projectile is dependent only upon the acceleration of gravity and not dependent upon the horizontal velocity. Thus, the vertical displacement (y) of a projectile can be predicted using the same equation used to find the displacement of a freefalling object undergoing onedimensional motion. This equation was discussed in Unit 1 of The Physics Classroom. The equation can be written as follows.






















Now consider displacement values for a projectile launched at an angle to the horizontal (i.e., a nonhorizontally launched projectile). How will the presence of a initial vertical component of velocity effect the values for the displacement? The diagram below depicts the position of a projectile launched at an angle to the horizontal. The projectile still falls 4.9 m, 19.6 m, 44.1 m, and 78.4 m below the straightline, gravityfree path. These distances are indicated on the diagram below.
The projectile still falls below its gravityfree path by a vertical distance of 0.5*g*t^2. However, the gravityfree path is no longer a horizontal line since the projectile is not launched horizontally. In the absence of gravity, a projectile would rise a vertical distance equivalent to the time multiplied by the vertical component of the initial velocity (v_{iy}• t). In the presence of gravity, it will fall a distance of 0.5 • g • t^{2}. Combining these two influences upon the vertical displacement yields the following equation.
where v_{iy} is the initial vertical velocity in m/s, t is the time in seconds, and g = 9.8 m/s/s (an approximate value of the acceleration of gravity). If a projectile is launched with an initial vertical velocity of 19.6 m/s and an initial horizontal velocity of 34.6 m/s, then the x and y displacements of the projectile can be calculated using the equations above. A sample calculation is shown below.

where v_{iy} = 19.6 m/s y = (19.6 m/s) * (1 s) + 0.5*(9.8 m/s/s)*(1 s)^{2} y = 19.6 m + (4.9 m) y = 14.7 m (approximately) 
where v_{ix} = 33.9 m/s x = (33.9 m/s) * (1 s) x = 33.9 m

The following table lists the results of such calculations for the first four seconds of the projectile's motion.


















The data in the table above show the symmetrical nature of a projectile's trajectory. The vertical displacement of a projectile t seconds before reaching the peak is the same as the vertical displacement of a projectile t seconds after reaching the peak. For example, the projectile reaches its peak at a time of 2 seconds; the vertical displacement is the same at 1 second (1 s before reaching the peak) is the same as it is at 3 seconds (1 s after reaching its peak). Furthermore, the time to reach the peak (2 seconds) is the same as the time to fall from its peak (2 seconds).
Use your understanding of projectiles to answer the following questions. Then click the button to view the answers.
1. Anna Litical drops a ball from rest from the top of 78.4meter high cliff. How much time will it take for the ball to reach the ground and at what height will the ball be after each second of motion?
2. A cannonball is launched horizontally from the top of an 78.4meter high cliff. How much time will it take for the ball to reach the ground and at what height will the ball be after each second of travel?
Click here to see a diagram of the situation.
3. Fill in the table below indicating the value of the horizontal and vertical components of velocity and acceleration for a projectile.
4. The diagram below shows the trajectory for a projectile launched nonhorizontally from an elevated position on top of a cliff. The initial horizontal and vertical components of the velocity are 8 m/s and 19.6 m/s respectively. Positions of the object at 1second intervals are shown. Determine the horizontal and vertical velocities at each instant shown in the diagram.
The following diagram pertains to questions #1 and #2 above. A scale is used where 1 cm = 5 meters. (Note that 1cm may be a different distance for different computer monitors; thus, a cmruler is given in the diagram.)
Lesson 2: Projectile Motion
 What is a Projectile?
 Characteristics of a Projectile's Trajectory
 Describing Projectiles with Numbers
 Horizontal and Vertical Components of Velocity
 Horizontal and Vertical Components of Velocity
 Initial Velocity Components
 Horizontally Launched Projectiles  ProblemSolving
 NonHorizontally Launched Projectiles  ProblemSolving
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19962007