1.2 Falling Balls
ball dropped from a short stool vs ladder
- a ball dropped from a tall ladder is more dangerous than the same ball dropped from a short stool. The farther the ball has to fall, the longer it takes to reach the ground and the more time it has to accelerate. -during its long fall from the tall ladder, the ball acquires a large downward velocity and becomes very hard to stop. If you try to catch it you have to exert a very large upward force on it to accelerate it upward and and bring it to rest quickly. Ex: you drop a marble from rest and after 1 second it has fallen downward a distance of 4.9 m. How far had it fa;;em after only .5 s? A: about 1.2, when you drop the marble from rest, it starts its descent slowly but picks up speed and covers the downward distance faster and faster. in the first .5 s, it travels only a quarter of the distance it travels in the first 1 s, or about 1.2 m.
Projectile motion: how does horizontal travel complicate the motion of a falling ball?
-Not much, you can often that an object's vertical motion independently of its horizontal motion--> separating vector quantities (acceleration, velocity, and position) into components, those portions of the quantities that lie along specific directions. -can completely specify the ball's position by its distance to right, its distance in front of you and its altitude above you. If its to your left, behind you or below you the corresponding components have negative values. -When you drop a ball from rest or toss it straight up, its motion is entirely vertical and only the upward components of its position, velocity and acceleration are important. -when moving horizontally, you'll need to pay attention to rightward and forward components of those vector quantities. -gravity only affects verticla motion, a bal can coast horizontally while falling vertically
Acceleration due to gravity
-downward vector, which on earth is 9.8 N/kg. =weight/mass -a ball falling near Earth's surface experiences a downward acceleration of 9.8 m/s^2 regardless of mass. -because falling objects at earth's surface accelerate downward at exactly the same rate, a billiard ball and bowling ball dropped from same height will reach the ground togehter.
Weigh
-exactly proportional to the object's mass. -mass and weight have very different attributes: weigh is how hard gravity pulls on a ball, and mass is how difficult that ball is to accelerate. -weight-mass x acceleration due to gravity.
Velocity of Falling Ball
-falling ball is one that has only its weight, the force due to gravity acting on it, and gravity causes any falling object to accelerate downward at a constant rate. -present velocity=initial velocity + acceleration x time--> vector quantities, motion is vertical, vectors point up or down. -for a ball falling from rest, the initial velocity is zero the acceleration down is 9.8 m/s^2 and the time you've been watching it is the time since it started to drop, after 1 second the ball has a downward velocity of 9.8 m/s, after 2 second it has a down velocity of 19.6. Its velocity continues to increase downward. Vectors that point down have negative means a velocity of a falling ball down for 2 sec is -19.6 m/s
Gravity
a physical phenomenon that produces attractive forces between every pair of objects in the universe. The only object massive enough and near enough to have obvious gravitational effects on us is our planet, Earth. Gravity weakens with distance, -Earth's gravity exerts a downward force on any object near its surface, that object is attracted directly toward the center of earth with a force called weight.
Ex.: if it takes you about 1.4 s to reach the water from the 10 meter fiving platform, how fast are you going just before you enter the water?
A: 14 m/s The downward acceleration due to gravity is 9.8 m/s^2. you fall for 1.4s during which time your velocity increases steadily in the downward direction. Since you start with 0 velocity the final velocity is 9.8m/s^2 x 1.4 s=13.72 m/s..
Ex: you drop a marble from rest, and after 1 s, its velocity is 9.8 m/s in the downward direction. what is your velocity after .5 second of falling?
A: 4.9 m/s in the downward direction. A freely falling object accelerates downward at a steady rate. Its velocity changes by 9.8 m/s in the downward direction each and every second. In half a seconds, the marble's velocity changes by only half that amount or 4.9 m/s
EX: if you want customers to have a sec free fall experience, how all will you need to build the tower from which they'll jump?
A: should be about 122 meters. As they fall the jumpers will travel downward at ever increasing speeds, since the jumpers start from rest and fall down for 5 sec use equation present position=initial position + initial velocity x time + 1/2 x acceleration x time^2 or final height = initial height - 1/2 x 9.8 m/s^2 x (5s)^2 =initial height -122.5 m The downward acceleration is indicated here by the negative change in height. At the end of 5 seconds, the jumpers will have fallen more than 122 m, and will be traveling downward at about 50 m/s. The tower will need additional height to slow the jumpers down and begin bouncing them back up, clearly a 5 sec fall is unrealistic.
Tossing the ball upward
-if the only force acting on an object is its weight, then the object is falling. However, a thrown ball is falling too, once it leaves your hand it's subject only to its weight and it accelerates down at 9.8 m/s ^2 -The equation present velocity = initial velocity + acceleration x time still describes how the ball's velocity depends on the fall time, but now the initial velocity isn't zero. If you toss the ball straight up in the air, it leaves your hand with a large upward velocity. As soon as you let go of the ball it begins to accelerate downward. If the ball's initial upward velocity is 29.4 m/s then after 1 sec its upward velocity is 19.6, after 2 second its only 9.8 m/s, and at a 3rd second the ball comes to a complete stop with a velocity of zero. It then descends from this peak heigh, falling just as it did when you dropped it. -the ball's flight before and after its peak is symmetrical. It travels upward quickly at first, since it has a large upward velocity. As its upward velocity diminishes,it travels more and more slowly until it comes to a stop. It begins to descen slow at first then faster and faster as it continues at constant downward acceleration. The time the the ball takes to rise from its initial position in your hand to its peak height is exactly equal to the time it takes to descend back down from that peak to you hand. -The larger the initial upward velocity of the ball, the longer it rises and the higher it goes before its velocity is reduced to zero. It then descends for the same amount of time it spent rising. The higher the ball goes before it begins to descends, the longer it takes to return to the ground and the faster its traveling when it arrives. That is why catching a high fly ball with your bare hands stings so much, the ball is traveling very, very fast when it hits your hand, and a large force is required to bring the ball to rest quick
the position of a falling ball
The ball's velocity continue to increases as it falls, but where exactly is the ball located? You need to know the ball's initial position-where it was when you started to watch it fall. -You can then describe the ball's present position in terms of its initial position, its initial velocity, its acceleration and the time that has passed since you started watching it. -You must use the ball's average velocity during the whole period you've been watching it. Since the ball's velocity has been changing uniformly from its initial velicty to its present velocity, the ball's average velocity is exactly halfway between the two individual velocities: average velocity=initial velocity+ 1/2 x acceleration x time. -the ball's present position differs from its initial position by this average velocity times the time that you have been watching it--> equation present position =initial position + initial velocity x time + 1/2 x acceleration x time^2 ex: the longer a stone has been falling, the more its heigh diminishes with each passing second. But it won't overtake a stone that was dropped next to it at an earler time or dropped from beneath at the same time. ***everything falls the same way, heavy or light, large or small, all objects take the same amount of time to fall a given distance
