Catapult Engineering Pilot Workshop

Transcription

1 Catapult Engineering Pilot Workshop LA Tech STEP Illustrations:

2 Some Background Info Galileo Galilei ( ) did experiments regarding Acceleration. He realized that the change in velocity of balls rolling down inclined planes and falling objects were accelerated by the same phenomenon and followed the same mathematical rules. Illustrations: 8

3 From rest Vf = adt Illustrations: Galileo observed that the final velocity of an object starting from rest and accelerating at a constant rate equals the product of the acceleration and the elapsed time. If it had an initial velocity, the final velocity will equal the sum of the initial velocity and the increase in velocity caused by the acceleration. w/ initial velocity Vf = Vo + adt

4 Objects fall toward Earth because of a force called gravity. Acceleration due to gravity (g) is 9.8 m/s 2 0 m/s 1 sec 9.8 m/s 2 sec 19.6 m/s 10 sec 98 m/s If a bowling ball dropped from the roof, after the 1 st second, it would be traveling 9.8 m/s. A second later its velocity is 19.6 m/s. After falling for 10 seconds, its velocity is 98 m/s or about 219 miles per hour! That s fast!

5 Isaac Newton ( ) Newton pondered Galileo s work and motion in general He realized that the force (gravity) that caused the acceleration noted by Galileo was the same force that kept the planets in their orbits Newton formulated three laws of motion Two are important right now for us

6 What exactly is a force? A force is a push or a pull A force can act though contact Spring, rope, chain, friction, etc A force can act a distance Gravity, magnetism, electrical Illustrations:

7 Newton s First Law The Law of Inertia Illustrations: An object at rest will remain at rest, or an object in motion will remain in motion with constant velocity when the net force acting on the object is zero

8 Newton s Second Law The Law of Acceleration The effect of an applied force is to accelerate a body in the direction of the force. The acceleration is proportional to the applied force and the mass of the object. F=ma

9 What does this have to do with a catapult? Hang on, I m getting there. Consider, A bullet is fired horizontally from a rifle A second bullet is dropped from the rifle s height at the exact instant the bullet leaves the rifle s barrel. Which bullet strikes the ground first? Justify your answer. Ignore Air resistance

10 What force is acting on the bullet flying horizontally? Which of Newton s Laws applies to this bullet? What force is acting on the bullet that was dropped? Which law applies to this one? How many components of motion are applied to the dropped bullet? How many components of motion are applied to the fired bullet?

11 That is right. They strike the ground at the same time. Why? Because the only force being applied to each of them in the vertical direction was gravity. Therefore they fell to the ground at the same rate. However, their flight paths (trajectories) are different. The bullet that was dropped had a path that was straight down. What kind of path did the other one follow?

12 Yes, A curved one. This is PROJECTILE MOTION. Look at the next slide for an animation of these concepts.

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14 A diagram showing the components of motion for the projectile launched with horizontal velocity, for example a fired bullet. What is Vyo?

15 Note, For Projectile Motion: In these illustrations there was an independence of horizontal and vertical motions. Horizontal motion is under Newton s first law; therefore, it is at constant horizontal velocity Vertical motion is under Newton s second law; therefore, it is at constant downwards acceleration The combination of these two motions results in the observed parabolic path of a projectile.

16 Now, lets launch the projectile at an upward angle. Again, What forces act vertically? Horizontally? As a result, what type of flight path is taken? What components of velocity are involved?

17 Diagram of a projectile launched at an upwards angle with an initial velocity of Vo.

18 Speed: V=Dd/Dt A few formulas The following acceleration formulas are based on or can be derived from Galileo's work: From rest Vf = adt w/ initial velocity Vf = Vo+ adt Dd = ½ adt 2 Dd = VoDt + ½ adt 2 Vf = 2aDd Vf = Vo 2 + 2aDd

19 How might this apply to a Catapult?

20 Projectile Motion (Motion in 2 Dimensions) Launch Angle ( ) v 0x v 0y height (h) distance (s), time (t) Oh, yeah. An object launched from a catapult is a projectile. It is launched with an initial velocity, Vo An initial horizontal velocity, Vox An initial vertical velocity, Voy

21 Launch Angle ( ) v 0x v 0y height (h) distance (s), time (t) A projectile is launched with an initial velocity of 22.0 m/s at an angle of 40.0 o. Calculate the range of the projectile. To calculate range, you need to use this formula: Ddx = VxDt Therefore, we need to calculate Dt, Vx But, to calculate Dt, we need to calculate Vy So let s get at it.

22 Launch Angle ( ) v 0x v 0y height (h) distance (s), time (t) First calculate horizontal and vertical components of Vo: Sin = Voy / Vo Voy = Vo Sin = 22.0 m/s x Sin 40.0 o = 14.1 m/s Cosin = Vox / Vo Vox = Vo cosin = 22.0 m/s x cosin 40.0 o = 16.9 m/s

23 Launch Angle ( ) v 0x v 0y height (h) Ddy distance (s), time (t) Now let s calculate Dt Ddy =VoyDt + ½ adt 2 = Dt (Voy + ½ adt) since projectile goes up and back down Ddy = 0 0 = Dt (Voy + ½ adt) 0 = (Voy + ½ adt) & 0 = Dt Dt =- [(2)(Voy)] / g a = g = -9.8m/s 2 = -[(2) (14.1 m/s)] /-9.8 m/s 2 =2.88 s flight time

24 Launch Angle ( ) v 0x v 0y height (h) Ddy distance (Ddx), time (t) Now we can finally calculate range. Range = Ddx Vox = Ddx / Dt Ddx = Vox Dt = (16.9 m/s)(2.88s) = m = 49 m

25 Wait How to you get the projectile up to it s initial velocity, Vo? Right, a force has to be applied to accelerate the projectile. That is where the spring comes in.

26 What s Next? In order to design and build a catapult to accomplish certain tasks, you are going to have to apply kinematic (motion) formulas and solve for the variables concerning projectile motion, angular acceleration, potential energy of springs, and other such stuff.. Fortunately for me, that is someone else s job to show you.

27 Thanks for that