An object has a constant acceleration of 40 ft/sec^2, 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.

d^2s/dt^2 = 40

ds/dt = 40 t + c
when t = 0, ds/dt = -20
so ds/dt = 40 t - 20
s = 20 t^2 -20 t + d
when t= 0, s = 10
so s = 20 t^2 - 20 t + 10

To find the position function, we need to integrate the acceleration function twice with respect to time.

Given:
Acceleration (a) = 40 ft/sec^2
Initial velocity (v0) = -20 ft/sec
Initial position (s0) = 10 ft

Integration of acceleration gives:
Velocity (v) = ∫a dt
= ∫40 dt
= 40t + C1

Where C1 is the constant of integration.

To find the value of C1, we can use the initial velocity. When t = 0, v = -20.
So, substituting t = 0 and v = -20 in the velocity equation:
-20 = 40(0) + C1
C1 = -20

Now, the equation for velocity becomes:
v = 40t - 20

Next, we integrate the velocity equation to find the position function.

Position (s) = ∫v dt
= ∫(40t - 20) dt
= 20t^2 - 20t + C2

Where C2 is the constant of integration.

To find the value of C2, we can use the initial position. When t = 0, s = 10.
So, substituting t = 0 and s = 10 in the position equation:
10 = 20(0)^2 - 20(0) + C2
C2 = 10

Finally, the position function (s(t)) becomes:
s(t) = 20t^2 - 20t + 10

Therefore, the position function describing the motion of the object is s(t) = 20t^2 - 20t + 10.

To find the position function, s(t), describing the motion of the object, you need to integrate the equation of acceleration with respect to time twice.

Given:
Acceleration, a = 40 ft/sec^2
Initial velocity, v₀ = -20 ft/sec
Initial position, s₀ = 10 ft

First, integrate the acceleration to find the velocity function, v(t):
∫ a dt = ∫ 40 dt = 40t + C₁

Next, use the initial velocity, v₀, to determine the constant of integration, C₁:
v(t) = 40t + C₁
v₀ = 40(0) + C₁
C₁ = -20

Now, we have the velocity function:
v(t) = 40t - 20 ft/sec

Finally, integrate the velocity function to find the position function, s(t):
∫ v(t) dt = ∫ (40t - 20) dt = 20t^2 - 20t + C₂

Use the initial position, s₀, to determine the constant of integration, C₂:
s(t) = 20t^2 - 20t + C₂
s₀ = 20(0)^2 - 20(0) + C₂
C₂ = 10

Therefore, the position function, s(t), describing the motion of the object is:
s(t) = 20t^2 - 20t + 10 ft