Horizontally Launched Projectiles

1. Horizontally Launched Projectiles A ball rolling of the table is an excellent example of an object thrown into the air with horizontal initial velocity (velocity at the time when the object is launched). The ball becomes airborne when leaving the table. If the air resistance and friction between the table and the ball can be neglected (and there are many situations in which that can be done – bowling for example) the ball moves with constant velocity before it becomes projectile. And this is exactly velocity with which it leaves the table. black vector represents initial velocity of the horizontally launched projectile

2. The ball would continue its motion with the same speed and direction unless there is an acceleration. In horizontal direction, when launched from the table, there is nothing that can give horizontal acceleration. Horizontal velocity remains constant, because we neglect air resistance in our calculations, and the ball covers equal distances in equal intervals of time. In vertical direction there is gravitational acceleration. The projectile accelerate downward. When these two motions are combined - vertical free fall motion and uniform horizontal motion - the trajectory will be a parabola. click me

3. u q ux If the initial velocity of the ball is at some angle toward horizontal the path of the ball will start upwards. uy • Horizontal component vx = uxis constant throughout the motion. Vertical component vy is decreasing on the way up, becoming zero at the top, and increasing on the way down. click me Horizontal component of motion for a projectile is completely independent of the vertical component of the motion. Their combined effects produce the variety of curved paths - parabolas that projectile follow.

4. Zookeeper who found the special way to feed monkey banana. If there was no gravity acting on either the banana or the monkey, banana moves in a straight line and the monkey does not fall once he lets go of the tree. As such, a banana aimed directly at the monkey will hit the monkey. The zookeeper aims at the monkey and shoots the banana very fast . The banana reaches the monkey before the monkey has fallen very far. The zookeeper aims above the monkey’s head.The banana misses the monkey, moving over his head. banana passes as far above the monkey's head as it was originally aimed. But there is no holiday for gravity The zookeeper aims at the monkey and shoots the banana with a slow speed . Banana hits the monkey after the monkey has fallen considerably far.

5. In conclusion, the key to the zookeeper's dilemma is to aim directly at the monkey. Both banana and monkey experience the same acceleration since gravity causes all objects to accelerate at the same rate regardless of their mass. Since both banana and monkey experience the same acceleration each will fall equal amounts below their gravity-free path. click me more monkeys

6. The horizontal and vertical components of the velocity of a cannonball launched at an angle As the ball rises towards its peak, it has a downward acceleration. As the cannonball falls, it has a downward acceleration. The downward acceleration changes the vertical component of velocity. In the absence of horizontal forces, there would be a constant velocity in the horizontal direction. The horizontal component of the velocity remains the same size throughout the entire motion of the cannonball. Remind your self continuously: forces are not required for an object to be moving; once in motion, the presence of forces will only serve to accelerate such objects.

7. Projectile motion is the combination of two independent motion in the same time interval: horizontal with constant velocity and free fall. After some time interval t the object is at the position P(x,y) Horizontal motion: vx = ux = u cos θ x = uxt P(x,y) y Vertical motion: uy = u sin θ vy = uy + gt (g = ±9.8 m/s2) y = uyt + (g/2) t2 x R (x = R,y = 0) Maximum height ymax vy = 0 Range R (x = R, y = 0 ) 0 = uyt + (g/2) t2 t = 2uy/g R = uxt = 2 uxuy/g R = 2 u2sin θcosθ The minimum speed occurs at the top of the parabola.

8. 750 R = 2 u2 sinθ cos θ the same range is obtained for two projection angles that add up to 900 600 450 ACTUAL PATH 300 150 IDEAL PATH Projectile thrown with the same speed at 300 and 600 will have the same range. The one at 300 remains in the air for a shorter time. Till now we neglected we either neglected or considered case in which air resistance was negligible. When the air resistance is significant, the range of projectile getting much smaller and the path is not true parabola. It is beyond the scope of this course to mathematically attack this problem. Air resistance is particularly significant for fast-moving objects. A batted baseball travels only about 60% as far in air as it would in a vacuum. The path is no longer parabola.

9. The Truck and The Ball Imagine a pickup truck moving with a constant speed along a city street. In the course of its motion, a ball is projected straight upwards by a launcher located in the bed of the truck. Imagine as well that the ball does not encounter a significant amount of air resistance. What will be the path of the ball and where will it be located with respect to the pickup truck? How can the motion of the ball be described? And where will the ball land with respect to the truck?

10. The balls are thrown horizontally. If there was no gravity the balls would continue horizontally for ever at the same speed. But one second after the balls are thrown, because of gravity, they had fallen 5 m below horizontal line no matter how fast they were thrown. If the second ball is thrown twice as fast it will go twice as far in the same time.. field If the ball were thrown three times as fast, it will go three times as far in the same time. Ten times as fast, ten times further. What if the ball were thrown so fast that the curvature of the earth came into play??? The ball follows a curved down path (parabola). But the Earth is curved too. What if the ball were thrown so fast that the curved path matched the Earth’s curve? If there were no air resistance (no slowing down and eventually hitting the Earth) the ball would orbit Earth. An Earth satellite (space station, communication satellite, scientific satellites,… ) is simply projectile traveling fast enough to fall around the Earth rather than into it.

11. Satellites in circular orbit, such as the moon or space station, fall beneath the paths they would follow if there were no gravity – straight line. During each second the moon travels about one km . In this distance it deviates about one millimeter from a straight line due to the earth’s gravitational pull (dotted line). The moon continually falls toward the earth, as do the planets around the sun. A satellite launched with speeds less than 8 km/s would eventually fall to the Earth. A satellite launched with a speed of 8 km/s would orbit the Earth in a circular path. Launched with a greater speed satellite would orbit the Earth in an elliptical path. If launched with too great of a speed, a satellite/projectile will escape Earth's gravitational influences and continue in motion without actually orbiting the Earth. Such a projectile will continue in motion until influenced by the gravitational influences of other celestial bodies. At 8 km/s atmospheric friction would melt a piece of iron (falling stars). Therefore satellites are launched to altitudes above 150 km. Don’t even try to think they are free of Earth’s gravity. The force of gravity at that altitude is almost as strong as it is at the surface. The only reason we put them there is that they are beyond Earth's atmosphere , not beyond Earth’s gravity.