An object has a constant acceleration of 40 ft/sec2, an initial velocity of -20 ft/sec, and an initial position of 10 ft. Find the position function, s(t), describing the motion of the object.

Question

 An object has a constant acceleration of 40 ft/sec², an initial velocity of -20 ft/sec, and an initial position of 10 ft. Find the position function, s(t), describing the motion of the object.

Answer 1

The relationship between acceleration, velocity and position can be summarized with the following equations:

v(t) = ∫a(t)dt + v(0)

s(t) = ∫v(t)dt + s(0)

Given, a(t) = 40 ft/sec², v(0) = -20 ft/sec and s(0) = 10 ft , we can solve by first solving for v(t), and then using v(t) to solve for s(t):

v(t) = ∫40dt - 20

v(t) = 40t - 20

s(t) = ∫(40t - 20)dt + 10

s(t) = 20t² - 20t + 10 ft

Having s(t), we can gain some intuition about the object's motion by plotting s(t).

s(t) is a parabola that opens up, with it's vertex at (0.5, 5).

At t=0 seconds, our object's position is decreasing.

At t=0.5 seconds, the object changes direction, and the position starts increasing forever