Question

Suppose that the speed v (in ft/s) of a skydiver t seconds after leaping from a plane is given by the equation 𝑣 = 190(1 − 𝑒 − 0.168𝑡).

(a) Graph 𝑣 versus 𝑡.

(b) By evaluating an appropriate limit, show that the graph of 𝑣 versus 𝑡 has a horizontal asymptote 𝑣 = 𝑐 for an appropriate constant 𝑐.

(c) What is the physical significance of the constant 𝑐 in part (b)?

Answer 1

(a) Graph 𝑣 versus 𝑡.

Once we graph this we find that there is an asymptote for v = c = 190.

(b) By evaluating an appropriate limit, show that the graph of 𝑣 versus 𝑡 has a horizontal asymptote 𝑣 = 𝑐 for an appropriate constant 𝑐.

We have to find the limit when $t\to \infin$ so

$\lim\limits_{x\, \to \infin} v=\lim\limits_{x\, \to \infin} 190(1-e^{-0.168t}) \\ \lim\limits_{x\, \to \infin} 190(1-e^{-0.168t})=190-\lim\limits_{x\, \to \infin} e^{-0.168t} \\ \lim\limits_{x\, \to \infin} 190(1-e^{-0.168t})=190$

(c) What is the physical significance of the constant 𝑐 in part (b)?

c = 190 is the velocity once $t\to \infin$, and this will be the limit speed expressed in ft/s.

Reference:

• Thomas, G. B., & Finney, R. L. (1961). Calculus. Addison-Wesley Publishing Company.