Linear and Quadratic Functions
and Modeling
The height s and vertical velocity v of an object
in free fall are given by
`
where t is time (in seconds), g ≈ 32 ft/sec2 ≈ 9.8 m/sec2
is the acceleration due to gravity, v0 is the initial vertical
velocity of the object, and s0 is its initial height.
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Linear and Quadratic Functions
and Modeling
Example: Use the data in the table to write models for the height and vertical velocity of the rubber ball. Then use these models to predict the maximum `
height of the ball and its vertical velocity when it hits the ground.
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Linear and Quadratic Functions
and Modeling
Example:
`
v(t) = ­9.532t + vo but we know vo = 3.7583
so the equation becomes
v(t) = ­9.532t + 3.7583
To find the maximum, we just use the find maximum function on the calculator and get that the maximum occurs at (0.402, 1.8).
This means that the highest point is 1.8 meters above ground when the time is .402 seconds.
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Linear and Quadratic Functions
and Modeling
Example:
To find the velocity when the ball hits the ground, we need to use the equation
v(t) = ­9.532t + 3.7583, but we need to find t, the time when the ball hits the ground. The ground is where s(t) = 0 (the height is zero), which means that all we need to do is find the zeroes of the equation s(t) = ­4.6764t2 + 3.7583t + 1.045.
We find the zeroes (when the ball is on the ground occur when t ≈ ­0.219 and
when t ≈ 1.0222. Clearly, nothing happens in negative time, so the only reasonable answer is t ≈ 1.0222. Using this, and the equation for velocity at time t, v(t) = ­9.532t + 3.7583, we can find v(1.0222) = ­9.532(1.0222) + 3.7583,
which equals approximately 5.99. This means that when the ball hits the ground, it is traveling at a speed of 5.99 m/s.
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