Dead End : The Physics of a Rollercoaster

Transcription

1 Dead End : The Physics of a Rollercoaster

2 OUR ROLLERCOASTER 27 seconds long! 7.5 meter long track! 8 points of analysis 1.57 meters Average velocity= distance/ time = 7.5m / 26 =0.288 meters per second 0.42 meters

3 What you need to know... Gravitational Potential Energy= stored energy due to object s position relative to the ground Kinetic Energy=energy of motion Momentum= a quantity defined as the product of an object s mass and velocity Angular Velocity= velocity of an object around a fixed axis Angular Acceleration= rate of change of angular velocity Centripetal Force= the net force directed toward the center of the circle Centripetal Acceleration= acc. of an object in circular motion Free Fall= an object that falls with the acceleration of gravity (9.81 m/s^2) Velocity= the speed of an object

4 Funnel =(.0118)/(7)=.0824 m/s the downhill before provides the initial velocity Snake curve.15 m.15 m diameter 7 seconds Rot. Avg velocity= theta/t KE= (½)mv^2 (½)(.006)(.0824)^2 =2.46 x 10^-8 Joules the kinetic energy is a result of the gravitational potential energy Centripetal Acc= Vt^2/ r.0824^2/.075=.0905 m/s^2 acceleration of object in circular motion is caused by changing direction of the marble the marble moves faster in the center since it has less distance to cover in the same time interval for one rotation Centripetal Force= (mass)(centripetal acc) =(.006)(.0905) =5.432 x 10^-4 N the centripetal force is necessary to keep the ball moving in a circular motion. the curve of the funnel provides this force.22 m long 1 second Avg velocity= d/t.22/1=.22 m/s.22m The velocity increases temporarily around the curve because of the change in direction KE= (½)mv^2 (½)(.006)(.22)^2 = x 10^-4 J Momentum P=mv P=.006x.22=.00132kg/m/s Conservation of energy energy conservation occurs even when acceleration varies although the speed of the marble varies with each curve, the energy remains constant due to the conservation of energy

5 Loop Angular velocity= theta/t.395 m S/r= theta s=.262 m r=.06 m.06 m Theta= 4.4 Time=.22 s The velocity is a result of the kinetic energy from the downhill. The velocity is great enough to allow the marble to be projected at the end of the loop. Angular Velocity= 20 rad/s Downhill Velocity=distance/time Distance=.395 meters Time=.41 seconds Velocity=.963 m/s KE=½(mv)^2 Mass=.006g Velocity=.963 m/s KE=.0028 J Acceleration= (vf-vi)/t vf=.963 m/s vi= 0 m/s t=.41s Acceleration= m/s^2 PEg= mgh Mass=.006g Gravity= 9.81 Height=.298 m PEg=.018 J Angular Acceleration= delta angular velocity/ delta time Delta angular velocity= 20 rad/s Delta time=.22 s Angular Acceleration= rad/s^2 Centripetal Acceleration= r(angular velocity)^2 Radius=.06 m Angular Velocity= 20 rad/s Centripetal Acceleration= 24 m/s^2 Centripetal Force= m(centripetal Acceleration) The gravitational potential energy is converted to kinetic energy, which Mass=.006 m can be seen as the marble rolls down the decline. There is enough Centripetal Acceleration= 24 m/s^2 kinetic energy created to allow the marble to successfully travel Centripetal Force=.144 N through the loop.

6 Uphill.12 m Spiral.05 m.62 m Delta Y=.05m Delta X=.12m Mass Marble=.006 KG t=.28s PEg = mgh=.006*9.81*.05=.0029 N V=d/t=.13/.28=.464 m/s Kinetic energy KE=½ *mv^2 = ½ *.006*.464^2= 6.458*10^-4 Momentum P=mv=.006*.464 = kg/m/s The momentum from earlier downhills gives the marble enough kinetic energy to travel through this uphill, then the kinetic energy changes to potential energy at the top which then converts to kinetic energy to travel through the rest of the rollercoaster ride..62 m long 3 spirals Radius=.05m t=1.48s M=.006 Kg V= d/t =.62/1.48 = Centripetal Acceleration Ac=vt^2/r =.419 * 1.48^2 /.05 = m/s Centripetal Force Fc= mv^2/ r = mvt^2/r =.110 N Kinetic energy KE= (½)mv^2 = ½ *.006 *.419^2 = *10^-4 N The Marble travels through the loop/jump and lands in the spiral, then it continues traveling, as it travels, it gains kinetic energy from the potential energy it had at the top and gaining speed for the marble to continue travel to the funnel and the more velocity it picks up, the more laps it will do in the funnel, resulting a long time spent there, even though each lap would be faster.

7 Projectile.055 m high.055 m jump The kinetic energy of the loop before it propels the marble in a parabolic path landing in the spiral. Last turns.055 m The projectile is in free fall (falling with the acceleration of gravity) with a horizontal velocity, which allows the marble to travel.055 meters The x and y components of the projectile are both about.055 meters, which means the angle at which the marble is propelled is about 45 degrees. The marble gains gravitational potential energy at the peak of its path. This energy is then converted to kinetic energy, allowing it to roll down the spiral following it..6 m long 2 seconds Avg V=d/t =(.6)(2)=.3m/s KE=(½)mv^2 (½)(.006)(.3^2) 0.6 m = J All the Gravitational potential energy created from the beginning has converted to kinetic energy, which then was converted to other forms of energy such as friction, heat, and sound. Since all the energy is no longer in the form of kinetic energy, the marble slows to a stop. Momentum= mv = (.006)(.3)=.0018 kg x m/s The curve of the turn forces the marble to change directions even though its inertia forces it to go straight. This causes the marble to hit the side of the track.

8 The Making of Dead End Details of construction -18 Columns -Construction score Total cost $ Final Score Beams -9 Diagonal Support -3 Merge -9 Straight tracks -5 Sharp turns -4 Wide turns -2 funnels -1 loop

9 The Spotlight Total elapse time: 27 seconds! over 1.5 meters tall! 7.5 meter long track! efficient use of space sturdy structure wide use of various components Full of excitement!