Automotive engineers refer to the time rate of change of acceleration as the "jerk." Assume an object moves in one dimension such that its jerk J is constant.

(a) Determine expressions for its acceleration ax(t), velocity vx(t), and position x(t), given that its initial acceleration, velocity, and position are ai, vi, and xi, respectively. (Use any variable or symbol stated above as necessary.)

ax(t) =

vx(t) =

x(t) =

(b) Show that ax2 = ai2 + 2J(vx − vi).
(Do this on paper. Your instructor may ask you to turn in this work.)

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(a) To determine the expressions for acceleration ax(t), velocity vx(t), and position x(t), we can use the kinematic equations integrating the jerk J.

Given initial acceleration, velocity, and position as ai, vi, and xi, respectively, we can write:

Acceleration:
ax(t) = ai + ∫ J dt
= ai + Jt + C₁

Velocity:
vx(t) = vi + ∫ ax(t) dt
= vi + ∫ (ai + Jt + C₁) dt
= vi + ait + (Jt²/2) + C₁t + C₂
= vi + ait + (Jt²/2) + Ct + C₂

Position:
x(t) = xi + ∫ vx(t) dt
= xi + ∫ (vi + ait + (Jt²/2) + Ct + C₂) dt
= xi + ∫ (vi + ait)dt + ∫ (Jt²/2)dt + ∫ (Ct)dt + ∫ (C₂)dt
= xi + vi(t) + (ai/2)t² + (Jt³/6) + (Ct²/2) + C₂t + C₃

where C₁, C₂, C₃ are constants of integration.

(b) To show that ax² = ai² + 2J(vx - vi), we need to substitute the expressions found in part (a) into the equation and simplify:

ax² = (ai + Jt + C₁)²

Expanding the equation:
ax² = (ai + Jt + C₁)(ai + Jt + C₁)
= ai² + Jait + C₁ai + Jai² + J²it² + JC₁t + C₁Jt + C₁Jt + C₁²
= ai² + 2Jait + 2C₁ai + J²t² + 2JC₁t + C₁²

Now substitute the expression for vx(t) - vi:

vx - vi = vi + ait + (Jt²/2) + Ct + C₂ - vi
= ait + (Jt²/2) + Ct + C₂

=ax² = ai² + 2J[(ait + (Jt²/2) + Ct + C₂) - vi]
= ai² + 2J(vx - vi)

Therefore, we have shown that ax² = ai² + 2J(vx - vi).

(a) To determine the expressions for acceleration ax(t), velocity vx(t), and position x(t), we can start by integrating the jerk J.

Since jerk is the time rate of change of acceleration, integrating J with respect to time gives us the acceleration:

ax(t) = ∫J dt

Since the jerk J is constant over time, we can represent it as a constant value J. Integrating a constant gives us:

ax(t) = Jt + C1

where C1 is the constant of integration. However, we are given the initial acceleration ai, which means at t = 0, ax(0) = ai. Substituting this condition into the equation, we can solve for the constant of integration:

ai = J(0) + C1
C1 = ai

Therefore, the expression for acceleration ax(t) is:

ax(t) = Jt + ai

Next, we can find the velocity by integrating the acceleration with respect to time:

vx(t) = ∫ax(t) dt
vx(t) = ∫(Jt + ai) dt

Integrating Jt gives us:

vx(t) = (J/2)t^2 + ait + C2

Again, we have an initial condition where vx(0) = vi. Substituting this condition into the equation, we can solve for the constant of integration:

vi = (J/2)(0)^2 + ai(0) + C2
C2 = vi

Therefore, the expression for velocity vx(t) is:

vx(t) = (J/2)t^2 + ait + vi

Finally, we can find the position by integrating the velocity with respect to time:

x(t) = ∫vx(t) dt
x(t) = ∫[(J/2)t^2 + ait + vi] dt

Integrating (J/2)t^2 gives us:

x(t) = (J/6)t^3 + (ai/2)t^2 + vit + C3

Using the initial position condition x(0) = xi, we can solve for the constant of integration:

xi = (J/6)(0)^3 + (ai/2)(0)^2 + vi(0) + C3
C3 = xi

Therefore, the expression for position x(t) is:

x(t) = (J/6)t^3 + (ai/2)t^2 + vit + xi

(b) To show that ax^2 = ai^2 + 2J(vx - vi), we substitute the expressions for acceleration and velocity:

ax^2 = (Jt + ai)^2
ax^2 = J^2t^2 + 2Jtai + ai^2

Then, substituting vx = (J/2)t^2 + ait + vi:

2J(vx - vi) = 2J[(J/2)t^2 + ait + vi - vi]
2J(vx - vi) = Jt^2 + 2Jait

Now, we can see that ax^2 = ai^2 + 2J(vx - vi) is satisfied, as the terms match in both equations:

ax^2 = J^2t^2 + 2Jtai + ai^2
ax^2 = Jt^2 + 2Jait + ai^2

Therefore, we have shown that ax^2 = ai^2 + 2J(vx - vi).