Projectile Motion

What is a Projectile? Anything that is thrown or shot through the air. Projectiles have velocities in two directions. Horizontal Motion: Motion parallel to the Earth’s surface. Vertical Motion: The force of gravity pulling down on the object.

Projectile Motion A projectile’s horizontal and vertical motion are completely independent of each other. Gravity will effect a projectile and a falling object in the same way. Therefore, if an object is dropped and thrown at the same time they will hit the ground at the same time. It does not matter that the projectile travels a farther distance.

Newton’s Cannon Newton described the possible trajectories of a cannonball shot from a tall mountain. Weaker shots fall in parabolas, but soon the curvature of the earth becomes more important. If you could ignore air resistance, stronger shots would clear the horizon. If you could launch the cannonball fast enough, its curve would match the curvature of the earth. This is called an orbit. A satellite is no more than a projectile moving fast enough to continually clear the horizon as it falls.

Curved Path A projectile is any object that is projected by some means and continues in motion by its own inertia. Without gravity, the object would follow a linear motion; with gravity, the path curves. The path is surprisingly simple if the horizontal and vertical components of motion are investigated separately.

HORIZONTAL COMPONENT: If friction is ignored (and we always ignore friction), then there are no forces acting on the x-axis. If net force is zero, then acceleration is zero. If acceleration is zero, then the object is either at rest or moving at constant velocity. Since we know that the object is already in motion, the projectile must be moving at constant speed on the x-axis. It moves of its own inertia and covers equal distances in equal intervals of time.

VERTICAL COMPONENT: Projectiles move just like a freely falling object along the y-axis. The changing motion is due only to acceleration due to gravity. On the way up, the object decreases its speed as it goes against gravity. On the way down, speed increases as it moves with gravity.

Projection Angles With the same initial speed but different projection angles, a projectile will reach different altitudes (height above the ground) and different ranges (distances traveled horizontally). However, the same range can be obtained from two different angles, symmetrically around a maximum of 45˚, as shown in the graph.

Symmetry The path of a projectile is symmetrical… It rises to its maximum height in the same time it takes to fall from that height to the ground. Because acceleration is the same all of the time, the speed it loses while going up is the same as the speed it gains while falling. Therefore the speeds are the same at equal distances from the maximum height, where the vertical speed is zero.

WHAT IF THE PROJECTILE IS SHOT UPWARD?

Now assume gravity is turned on! The projectile would travel with a parabolic trajectory. The downward force of gravity will act upon the cannonball to cause the same vertical motion as before - a downward acceleration. The cannonball falls the same amount of distance in every second as it did when it was merely dropped from rest.

REVIEW TO THIS POINT! a projectile is any object upon which the only force acting upon it is gravity, projectiles travel with a parabolic trajectory due to the influence of gravity, there are no horizontal forces acting upon projectiles and thus no horizontal acceleration, the horizontal velocity of a projectile is constant (never changing in value), there is a vertical acceleration caused by gravity; its value is -9.8 m/s/s, down, the vertical velocity of a projectile changes by 9.8 m/s each second, the horizontal motion of a projectile is independent of its vertical motion.

PROJECTILE MOTION EQUATIONS The following are the formulas you will need to solve projectile motion problems. They are derived from the kinematic equations. y = Vyt + 1/2gt2 x = Vxt Vy = Vyo + gt

Example (Horizontally Launched Projectile): A steel ball rolls with a constant velocity across a table top 0.598m high. The steel ball rolls towards the edge and falls to the ground. When it eventually hits the ground, it had travelled 0.852m horizontally from the edge of the table. How fast was the ball rolling?

Solution: Given: Unknown: Vx = ? x = 0.852 m Y = - 0.598 m Vyo = 0 m/s Equations: x = Vxt y = vyt + ½ gt2 Vy = Vyo + gt

Solution (Cont’d): y = Vyt + ½ gt2 y = ½ gt2 (Vy = 0 m/s) t2 = 2y/g t = √(2y/g) t = √[(2)(-0.598 m) / (-9.8 m/s2) t = 0.34 s

Solution (Cont’d): Therefore: x = Vxt Vx = x/t Vx = 0.852 m / 0.34 s

LAUNCHED AT AN ANGLE What if we have an object that is launched at an angle and lands at the same height as it was launched?

EXAMPLE PROBLEM Bubba Watson crushes another drive off the tee at The Masters golf tournament. He drives the ball at a 30° angle with a velocity of 38 m/s. a.) How long does it take for the golf ball to reach its maximum height? b.) What is the maximum height? c.) What is the length of the drive?

SOLUTION