Projectile Motion Analyzed
Now we are ready to analyze projectile motion, horizontally and vertically.
The Horizontal Motion
Because there is no acceleration in the horizontal direction, the horizontal component of the projectile’s velocity remains unchanged from its initial value throughout the motion as demonstrated in . At any time the projectile’s horizontal displacement from an initial position is given by which we write as
The Vertical Motion
The vertical motion is the motion we discussed in for a particle in free fall. Most important is that the acceleration is constant. Thus, the equations of provided we substitute -g for a and switch to y notation.
and its magnitude steadily decreases to zero, which marks the maximum height of the path. The vertical velocity component then reverses direction, and its magnitude becomes larger with time.
The Equation of the Path
We can find the equation of the projectile’s path (its trajectory) by eliminating t between for t and substituting into, we obtain, after a little rearrangement.
This is the equation of the path shown in . In deriving it, for simplicity respectively. Because constants of the form which a and b are constants. This is the equation of a parabola, so the path is parabolic.
The Horizontal Range
The horizontal range R of the projectile as shows, is the horizontal distance the projectile has traveled when it returns to its initial (launch) height. To find range R, let us put x and y, obtaining
Caution: This equation does not give the horizontal distance traveled by a projectile when the final height is not the launch height.
The horizontal range R is maximum for a launch angle of 45°.
The Effects Of the Air
We have assumed that the air through which the projectile moves has no effect on its motion. However, in many situations, the disagreement between our calculations and the actual motion of the projectile can be large because the air resists (or opposes) the motion,for example, shows two paths for a fly ball that leaves the
bat at an angle of 60° with the horizontal and an initial speed . Path I (the baseball player’s fly ball) is a calculated path that approximates normal conditions of play in air. Path (the physics professor’s fly ball) is the path that the ball would follow in a vacuum.
CHECKPOINT 5: A fly ball is hit to the outfield. During its flight (ignore the effects of the air), what happens to its (a) horizontal and (b) vertical components of velocity? What are the (c) horizontal and (d) vertical components of its acceleration during its ascent and its descent, and at the topmost point of its flight?