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AP50 Fall 2014 Estimation Activity 3 Solutions Th Oct 9 1) Estimate the chemical energy converted to kinetic energy during the explosive separation of a professional fireworks shell. Answer units: [J] Getting started Think of the exploding shell as converting stored chemical energy to kinetic energy. Assuming that all pieces of the shell fly apart at the same speed, we can calculate the energy released. Devise plan Execute plan Estimate for m: 5kg Estimate for v: Assumption: Fireworks explode at approximately 100m height Assumption: The radius of the explosion is approximately 1/10 the height (10m) Assumption: The time for an ignited particle to reach its maximum radius is about 0.5s. E = 103 J order of magnitude = 3 Evaluate plan Some fireworks are carbon based, meaning they are composed of gunpowder and charcoal. We can check the reasonability of our answer by comparing it to the chemical energy converted to kinetic energy for a shell leaving the muzzle of a gun, since this reaction is also relies on chemical reactions involving gunpowder. We can estimate the mass of the shell as 200g and the muzzle velocity as 100 m/s. The kinetic energy of the shell at the muzzle is then K=(1/2)mv2=(1/2)(0.2kg)(100 m/s)2=103 J, which is the same kinetic energy we estimated for a firework explosion. 2) Estimate the relative speed of the moon as seen from an observer on the earth. Answer units: [m/s] Getting started We know that the earth rotates while the moon orbits, and that the difference between these two rates will give us relative velocity. We can therefore subtract the velocity of the moon from the velocity of an observer on earth’s surface. Linear velocities can be estimated using arc length formula. Devise plan AP50 Fall 2014 Execute plan Assumption: Radius of earth = 6000km Assumption: Radius of moon orbit = 380,000km Assumption: Moon travels around earth in about 30 days Assumption: 1 earth rotation is 24 hours Vme = -9.2 * 102 m/s order of magnitude: 3 Evaluate plan To quickly estimate how reasonable this result is, we can consider that it takes the moon 30 days to complete an orbit that is about 100 times larger than the rotation completed by the Earth in a day. Thus we can compare the 100𝑥 𝑥 70𝑥 velocity of each as versus , we can simply calculate that the difference is approximately or a bit more than 30 1 30 twice the original distance – 14000km traveled in a day. 14000 km per day translates to approximately 200 m/s which is on the same order of magnitude as our calculated result. Alternatively, we can consider that the moon has to travel a distance 100 times larger than the Earth within a given period, which translates directly to the relative difference in velocity. 3) Estimate the power output by an average person completing a “stadium workout” where they run up and down each section of the Harvard Stadium as fast as they can. Answer units: [W] Getting started Since the workout requires running up stadium sections, we can estimate the height climbed over the course of the entire workout. Given a person’s weight, potential energy gain can be calculated. Estimating the time it would take to complete the stadium run would provide all information necessary to estimate work. Devise plan AP50 Fall 2014 Execute plan Assumption: m =70kg Assumption: h = 600m. There are about 40 sections measuring about 15m high each, which multiplies to a total of 600m climbed. Assumption: t =30 minutes 𝑃 ≈ 1.3 ∗ 102 𝑊 order of magnitude = 2 Evaluate plan The same as a light bulb! This is reasonable because we know that some bikes have generator lights, and its possible for a person to power the lights by pedaling. 4) Estimate the force exerted by the atmosphere on the entire surface of one person’s skin. Answer units: [N] Getting started Pressure = force/area, and we know atmospheric pressure is approximately 100,000Pa. We just need to estimate the surface area of one human’s skin. Devise plan Execute plan Assume A = 1-9 m2 𝐹 = 105 𝑁 order of magnitude = 5 Evaluate plan We can find that an average adult has a skin surface area of 2 square meters, which is very close to the assumed area. The calculated force given known atmospheric pressure is fairly straightforward and the large force is reasonable when considering the force exerted by all of the molecules surrounding the skin in the atmosphere. 5) Estimate the height of the mountain a college student could climb with the energy saved by turning out their dorm lights instead of keeping the lights on for an all-nighter homework assignment. Answer units: [m] Getting started Estimate the energy required to run a light bulb overnight, and then convert to potential energy to determine height climbed. Devise plan AP50 Fall 2014 Execute plan Assumption: Light bulb burns for 12 hours Assumption: Light bulb is 100W Assumption: Person is 70kg ℎ = 6.3 ∗ 103 𝑚 order of magnitude = 4 Evaluate plan We consider that a light bulb uses a significant amount of electrical power if used over the course of an entire night. The calculation for the bulb usage in a single night is fairly straightforward given the translation between watts and joules. To calculate the height a college student could climb using this energy, we consider only the corresponding potential energy (for simplicity) and neglect other energy that might be needed to climb a height almost equivalent to Mount Everest.