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---------------------------- - 2293 - Work gets you Olympic Awards
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- This is an exercise for you to try own your own. Let’s see if we can get the same answers. You are the judge for the Olympic trials in field events. You have gold, silver, and bronze medals to give to the three best athletes. And, you only have three athletes in these events. Unfortunately, they are not competing in the same event:
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--------------------------------- The shot putter throws the 16 pound shot 63.3 feet.
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--------------------------------- The discus thrower throws the 4.4 pound discuss 196.4 feet
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--------------------------------- The javelin thrower throws the 1.76 pound javelin 282.4 feet.
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- Obviously you want to give the awards to the best athletes. Should it be the athlete who threw the farthest? Or, the athlete you threw the heaviest weight? Which one did the most work? Who gets the gold
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- Work is the key word in physics. Work is the change in Total Energy.
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- Total Energy = Kinetic Energy plus Potential Energy plus Thermal Energy. Total Energy = the energy in motion with the mass times the square of the velocity of the projectile, plus the potential energy of the force of gravity that limit’s the height the projectile will reach, plus the energy to overcome the friction of air resistance, which is a form of thermal energy.
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- The Law of Conversation of Energy requires the Total Energy to remain constant, the projectile only changes its form of energy in flight, until it reaches its peak elevation, and until it hit’s the ground, which is again thermal energy.
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- We will neglect thermal energy and the affect of air resistance, it’s small anyway.
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---------------------------- Total Energy = Kinetic Energy + Potential Energy.
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---------------------------- E = K.E. + P.E.
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---------------------------- E = ½ m* v^2 + mgh
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---------------------------- where: ‘m’ is the mass, ‘v’ is the velocity, ‘g’ is the acceleration of gravity, and ‘h’ is the height above ground. “g” is 32 feet/second/second, or 9.8 meters /second^2.
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- To start with the shot putter, converting to metric units, the 7.25 kilogram shot went 19.3 meters.
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--------------------------------------- E = ½*7.25* v^2 + 7.25*9.8* h
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- Unfortunately we do not know the velocity, “v” and we do not know the height, “h“. We do know that a projectile reaches its maximum distance when it is thrown at an angle of 45 degrees ( neglecting air resistance.) We also know that with the constant acceleration of gravity the trajectory of the projectile will be parabolic.
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- The initial velocity launched at 45 degrees will have two components. The y-direction, or vertical component, will be the sine of 45 or 0.707 * velocity. The x-direction, or horizontal component, will be the cosine of 45 or 0.707* velocity.
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- The x-direction will be a constant velocity because there is no air resistance to slow it down. The y-direction will always be fighting the pull of gravity and its velocity will slow until it reaches a maximum height and at its peak the y-direction velocity becomes zero and reverses in its downward path to the ground.
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- The general formula for the distance thrown is the average velocity times time.
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- Since the velocity in the x-direction is constant it is the average. There is no affect of acceleration in the x-direction. For the shot putter:
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--------------------------------------------------- distance = velocity * time
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--------------------------------------------------- Vx * t = 19.3 meters
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--------------------------------------------------- Vx = 0.707 v
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--------------------------------------------------- time = t = 19.3 / 0.707*v
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- The velocity in the y-direction is affected by the acceleration of gravity, g = -9.8 m/sec^2. The velocity , “Vy” changes during flight. Because this is constant acceleration the velocity can be calculated as the change in instantaneous velocity to original velocity divided by the change in time.
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- The average velocity is the initial velocity plus the velocity at time “t” divided by 2. The acceleration is the average velocity per unit time. The acceleration is equal to the change in velocity per the change in time.
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---------------------------------------------- acceleration average = (Vi + v) / t - to
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--------------------------------------------- where: Average acceleration = g = 9.8 m/sec^2
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-------------------------------------------- where: Vi = instantaneous velocity
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-------------------------------------------- where: v = initial velocity
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-------------------------------------------- where: to = time zero = 0
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- Solving for the instantaneous velocity we get the general formula for velocity with constant acceleration,
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---------------------------------------------- Vi = at + v:
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---------------------------------------------- Vi = g t + Vy
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- Under constant acceleration the average velocity can be calculated to be the sum of the two velocities divided by 2:
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---------------------------------------------- Vy average = (Vi + Vy)/2
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- And, the distance can be calculated to be average velocity * time:
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--------------------------------------------- y- distance = (Vi + Vy)* t /2
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- Now, substituting in the equation for the instantaneous velocity we get the general formula for distance with constant acceleration,
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---------------------------------------------- distance = 1/2a*t + v*t
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---------------------------------------------- y- distance = (g t + Vy + Vy)* t /2
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--------------------------------------------- y- distance = ½*g t^2 + Vy*t
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- Next, we know that at mid-flight, half the total time of flight, and, half the x-distance, the y-distance is maximum and the y-direction velocity is zero.
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------------------------------------ Vi = g t + Vy
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---------------------------------- at t/2 Vi = zero
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--------------------------------- Vy = - g t/2
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---------------------------------- 0.707 v = 9.8 * 19.3 / .707v / 2
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---------------------------------- v^2 = 189
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--------------------------------- v = 13.75 meters / second
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-------------------------------- K.E. = ½ m*v^2
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--------------------------------- K.E. = ½ * 7.25 * 13.75^2
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---------------------------------- K.E. = 686 kg m^2/sec^2
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- kg*m^2/sec^2 are the units of energy called “joules“. Actually, the kinetic energy was a little bit less because the athletes were 6 foot tall and released their projectiles above the ground. You need to subtract 1.8 meters from their distances. K.E. is then 625 joules for the shot putter.
- The Kinetic Energy provided by the discus thrower , where the mass of the discus is 2 kilograms and the distance thrown is 59.87 meters.
------------------------------- t = 59.87 / Vx
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-------------------------------- Vy = 9.8*t/2
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------------------------------- .707 v = 9.8 * 59.87/ .707 v / 2
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------------------------------- v = 24.2 m/sec
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------------------------------- K.E. = 587 joules
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- The kinetic Energy provided by the javelin thrower when the mass of the javelin is 0.8 kilograms and the distance thrown is 86.09 meters.
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------------------------------- t = 86.09 / Vx
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-------------------------------- Vy = 9.8*t/2
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------------------------------- .707 v = 9.8 * 86.09/ .707 v / 2
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------------------------------- v = 29 m/sec
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------------------------------- K.E. = 337 joules
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- You are ready to present the awards:
------------------- GOLD -------------------- SILVER ------------- BRONZE
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------------------ Shotput -------------------- Discus --------------- Javelin
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“m“ ----------- 16 pounds --------------------- 4.4--------------------- 1.76
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“v“ ------------- 31 miles / hour ---------------- 54 ---------------- 64
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“x“ ------------ 63 feet ------------------------- 196 ---------------- 282
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K.E.------------ 686 joules -------------------- 587 ---------------- 337
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- I do not know if the ground feels warm after a 16 pound shotput hits at 31 miles per hour.
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- But, an asteroid of 4,600,000,000 tons hitting at 67,000 miles per hour is 4*10^21 joules of thermal energy. That has got to feel hot.
- Here is energy put in perspective:
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-------------------------------- Falling raindrop --------------------------------- 0.001 joules
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------------------------------- Baseball pitch ---------------------------------- 100
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-------------------------------- Our shotputter --------------------------------- 686
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-------------------------------- A running human ----------------------------- 1,000
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-------------------------------- An automobile on highway -----------------1,000,000
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-------------------------------- Your daily food intake ---------------------10,000,000
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--------------------------------- One megaton of TNT --------------------- 4.2*10^15
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-------------------------------- U.S. Energy consumed/year ------------ 10^20
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-------------------------------- The asteroid impact ----------------------- 10^21
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---------------------------------- Energy from the Sun -------------------- 10^34
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- March 4, 2019. 894
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--------------------- Tuesday, March 5, 2019 -------------------------
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