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A skydiver jumps from an airplane and accelerates toward the ground. His velocity can be modeled by the function

v(t) = 1000t/5t+8, where v (t) is the velocity in feet per second, and t represents time in seconds. Due to air resistance, there is

a limiting velocity the skydiver will not exceed, called the "terminal velocity". At terminal velocity, the velocity does not

continue to increase. What is the terminal velocity? Justify your answer.

Sagot :

MrRoyal

Answer:

The terminal velocity is 8.00ft/s

Step-by-step explanation:

Given

[tex]v(t) = 1000t^2 - 5t + 8[/tex]

Required

The terminal velocity

This implies that we calculate the maximum velocity.

First, we calculate the maximum value of t using:

[tex]t_{max} = -\frac{b}{2a}[/tex]

Where:

[tex]v(t) = at^2 + bt + c[/tex]

So, we have:

[tex]t_{max} = -\frac{-5}{2*1000}[/tex]

[tex]t_{max} = \frac{5}{2000}[/tex]

[tex]t_{max} = 0.0025[/tex]

Substitute this value of t in [tex]v(t) = 1000t^2 - 5t + 8[/tex] to get the maximum velocity

[tex]v(t) = 1000t^2 - 5t + 8[/tex]

[tex]v(t) = 1000 * 0.0025^2 - 5 * 0.0025 + 8[/tex]

Using a calculator, we have:

[tex]v(t) = 7.99375[/tex]

Approximate

[tex]v_{max} = 8.00ft/s[/tex]

Velocity is the rate of change of its position with respect to time. The terminal velocity of the skydiver is 200 ft/sec.

What is velocity?

Velocity is the rate of change of its position with respect to time.

[tex]V = \dfrac{dy}{dt}[/tex]

Given the velocity of the skydiver by the function v(t)=1000t/(5t+8), after some time, the velocity will become terminal velocity and then it can not be increased further, therefore, the terminal velocity can be written as,

[tex]\lim_{t \to \infty} V(t) = \lim_{t \to \infty} \dfrac{1000t}{5t+8}\\\\[/tex]

                    [tex]= \lim_{t \to \infty} \dfrac{\frac{1000t}{t}}{\frac{5t}{t}+\frac{8}{t}}\\\\[/tex]

                    [tex]= \dfrac{1000}{5+\frac{8}{t}}\\\\= \dfrac{1000}{5}\\\\=200[/tex]

Hence, the terminal velocity of the skydiver is 200 ft/sec. It can be confirmed by plotting the function on the graph.

Learn more about the Terminal Velocity:

https://brainly.com/question/18084516

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