ax = a cos ș
ay = a sin ș
ș = angle with positive x axis
ax = x component of acceleration
ay = y component of acceleration
4.3 - Projectile motion
Projectile motion: Movement determined by an
object’s initial velocity and the constant
acceleration of gravity.
The path of a cannonball provides a classic example of projectile motion.
The cannonball leaves the cannon with an initial velocity and, ignoring air resistance,
that initial velocity changes during the flight of the cannonball due solely to the
acceleration due to the Earth’s gravity.
The cannon shown in the illustrations fires the ball horizontally. After the initial blast, the
cannon no longer exerts any force on the cannonball. The cannonball’s horizontal
velocity does not change until it hits the ground.
Once the cannonball begins its flight, the force of gravity accelerates it toward the
ground. The force of gravity does not alter the cannonball’s horizontal velocity; it only
affects its vertical velocity, accelerating the cannonball toward the ground. Its y velocity has an increasingly negative value as it moves through
the air.
The time it takes for the cannonball to hit the ground is completely unaffected by its horizontal velocity. It makes no difference if the cannonball
flies out of the cannon with a horizontal velocity of 300 m/s or if it drops out of the cannon’s mouth with a horizontal velocity of 0 m/s. In either
case, the cannonball will take the same amount of time to land. Its vertical motion is determined solely by the acceleration due to gravity,
í9.80 m/s^2.
The equations to the right illustrate how you can use the x and y components of the cannonball’s velocity and acceleration to determine how
long it will take to reach the ground (its flight time) and how far it will travel horizontally (its range). These are standard motion equations
applied in the x and y directions. They hold true when the acceleration is constant, as is the case with projectile motion, where the acceleration
along each dimension is constant.
In Equation 1, we show how to determine how long it takes a projectile to reach the ground. This equation holds true when the initial vertical
velocity is zero, as it is when a cannon fires horizontally. To derive the equation, we use a standard linear motion equation applied to the
vertical, or y, dimension. You see that equation in the second line in Equation 1. We substitute zero for the initial vertical velocity and then
solve for t, the elapsed time.
Projectile velocity components
x and y velocity components
·x velocity constant (ax = 0)
·yvelocity changes(ay=í9.80 m/s^2 )Projectile motion
Motion in one dimension independent of
motion in other