4.10 - Projectile motion: aiming a cannon
The complexities of correctly aiming artillery pieces have challenged leaders as famed
as Napoleon and President Harry S. Truman. Because a cannon typically fires
projectiles at a particular speed, aiming the cannon to hit a target downfield involves
adjusting the cannon’s angle relative to the ground. If you break the motion into
components, you can determine how far a projectile with a given speed and angle will
travel.
To determine when and where a cannonball will land, you must consider horizontal and
vertical motion separately. To start, convert its initial speed and angle into x and y
velocity components. The horizontal velocity will equal the initial speed of the ball
multiplied by the cosine of the angle at which the cannonball is launched. The horizontal
velocity will not change as the cannonball flies toward the target.
The initial y velocity equals the initial speed times the sine of the launch angle. The y
velocity is not constant. It changes at the rate of í9.80 m/s^2. When the cannonball lands
at the same height at which it was fired, its final y velocity is equal but opposite to its
initial y velocity.
The initial y velocity of the projectile determines how long it stays in the air. As
mentioned, the cannonball lands with a final y velocity equal to the negative of the initial
y velocity, that is, vyf = ívyi. This means the change in y velocity equals í 2 vyi.
Knowingthis, and the value for acceleration due to gravity, enables us to rearrange a
standardmotion equation (vf = vi + at) and solve for the elapsed time. The equation for
the flight time of a projectile is the third one in Equation 1.
Once you know how long the ball stays in the air, you can determine how far it travels
by multiplying the horizontal velocity by the ball’s flight time. This is the final equation on
the right.
You can use these equations to solve projectile motion problems, but understanding the
analysis that led to the equations is more important than knowing the equations. Recall
the basic principles of projectile motion: The x velocity is constant, the y velocity
changes at the rate of í9.80 m/s^2 , and the projectile’s final y velocity is the opposite of
its initial y velocity.
Aiming a projectile, step 1
Start with initial angle, speed
Separate into x and y componentsAiming a projectile, step 2
Use initial y velocity and ay to calculate
flight time
Use initial x velocity and flight time to
calculate range