Other Applications of Velocity, Acceleration 3.2.

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Presentation transcript:

Other Applications of Velocity, Acceleration 3.2

Free Falling Objects Gravity -32 for feet -9.8 for meters Time Initial Velocity Initial Position

A dynamite blast blows a heavy rock straight up with a launch velocity of 160 ft/sec. How high does the rock go? First we need s(t) Highest point is the maximum. This occurs when v = 0 What are the velocity and speed of the rock when it is 256 ft. above the ground on the way up? On the way down? We want velocity when s(t)=256 Speed is the absolute value of velocity, so the speed is 96 ft/sec

What is the acceleration at any time t during its flight? The acceleration is always -32 When does the rock hit the ground again? Rock hits when s(t)=0

Free Fall A heavy ball bearing is released from rest at time t = 0 seconds How many meters does the ball fall in the first 2 seconds? Initial velocity is 0 since we are starting from rest. Initial height is 0 since we are tracking the distance the object falls What is the velocity, speed and acceleration at this time? Speed = 19.6

Other Applications The number of gallons of water in a tank t minutes after the tank has started to drain is: How fast is the water running out at the end of 10 minutes? We want Q’(t) when t = 10 What is the average rate at which the water flows out during the first 10 minutes? Average rate of change is slope

Economics Application In manufacturing, cost of producing widgets is c(x) where x is the number of units produced. Marginal Cost: rate of change of cost wrt level of production

Economics Application Suppose that is costs: dollars to produce x radiators when 8 to 30 radiators are produced and that gives the dollar revenue from selling x radiators. About how much extra will it cost to produce one more radiator a day, given you produce 10 now and what is your estimated increase in revenue for selling 11 radiators a day? The cost of producing 1 more radiator is the marginal cost at x = 10 Marginal Cost Marginal revenue at x = 10 will give us the estimated increase in revenue!