Question

Falling Bodies. In the simplest model of the motion of a falling body, the velocity increases in proportion to the increase in the time that the body has been falling. If the velocity is given in feet per second, measurements show the constant of proportionality is approximately

32. a) A ball is falling at a velocity of 40 feet/sec after 1 second. How fast is it falling after 3 seconds?

b) Express the change in the ball’s velocity ∆v as a linear function of the change in time ∆t.

c) Express v as a linear function of t. The model can be expanded to keep track of the distance that the body has fallen. If the distance d is measured in feet, the units of d ′ are feet per second; in fact, d ′ = v. So the model describing the motion of the body is given by the rate equations d ′ = v feet per second; v ′ = 32 feet per second per second.

d) At what rate is the distance increasing after 1 second? After 2 seconds? After 3 seconds?

e) Is d a linear function of t? Explain your answer.

Answer 1

a)

b)

where is change in velocity and is change in time and k is proportionality constant

c)

d)

e) no, d is not a linear function of t but quadratic fuction of t. since we solved in part c,

d = 8t + 16t²

so d is quadratic in t.