Notes and Explanations Projectile Motion Notes and Explanations Some diagrams and simulations from www.physicsclassroom.com

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.

Simulation of 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. Simulation of Newton’s Cannon

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.

Vector Combination The curved path of a projectile is a combination of these horizontal and vertical components. The curved path of a ball (as shown in the diagram) is best analyzed by considering the horizontal and vertical components of motion separately. Note that the ball’s horizontal component of motion doesn’t change as the falling ball moves forward. Note the vertical positions become farther apart with time and that the vertical distances traveled are the same as if the ball were simply dropped.

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.

Questions: At the instant a horizontally held rifle is fired over level ground, a bullet held at the side of the rifle is released and drops to the ground. Ignoring air resistance, which bullet strikes the ground first? A projectile is launched at an angle into the air. If air resistance is negligible, what is the acceleration of its vertical component of motion? Of its horizontal component of motion? At what part of its trajectory does a projectile have minimum speed? A ball tossed into the air undergoes acceleration while it follows a parabolic path. When the sun is directly overhead, does the shadow of the ball across the field also accelerate? Draw a free body diagram of a projectile. What type of motion occurs on the horizontal axis? How in the world can object be moving upward if the only force acting upon it is gravity? Are horizontal forces required to keep the projectile moving horizontally?

Answers They strike at the same time. -9.8 m/s2; 0 At maximum height. NO! Should show weight only. Constant Inertia

Memorize the following conceptual notions about projectiles: a projectile is any object upon which the only force 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 ~10 m/s2, down, the vertical velocity of a projectile changes by ~10 m/s each second, the horizontal motion of a projectile is independent of its vertical motion. the speed of a projectile is symmetrical around its maximum height.

Use these sites to answer the questions on the next slide. Simulation #1 Simulation #3 Simulation #2 Simulation #4

Questions What launch angle maximizes the range (horizontal distance)? What launch angle maximizes the height reached? How does the time spent in the air depend on the launch angle? Compare what happens at one angle, and then at 90 degrees minus that angle (30 and 60, for instance). What does Newton’s Cannon show? What happens to the trajectory when the mass of the projectile is changed? What effect does air resistance have on a trajectory? What happens to the trajectory when the initial velocity is changed? What conditions must exist so that two projectiles will collide in midair? What effect do the initial velocities, height differences, and angles have on this collision? Compare the trajectories of a projectile that is under the influence of gravity and one that is not?