Automotive engineers refer to the time rate of change of...

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 $a_{x}(t),$ velocity $v_{x}(t),$ and position $x(t),$ given that its initial acceleration, velocity, and position are $a_{x j}, v_{x i},$ and $x_{i},$ respectively. (b) Show that $a_{x}^{2}=a_{x i}^{2}+2 J\left(v_{x}-v_{x i}\right) .$

Key Concepts

Jerk

Jerk is the rate of change of acceleration with respect to time. In problems involving constant jerk, the acceleration changes linearly over time. This concept extends the basic ideas of kinematics, where constant acceleration leads to linear velocity change and quadratic displacement change, to a case where even the acceleration is not constant but changes uniformly.

Integration in Kinematics

Finding expressions for acceleration, velocity, and position involves successive integration. Starting with a constant jerk, one integrates to obtain the acceleration function, then integrates acceleration to yield the velocity function, and finally integrates velocity to determine the displacement. Each integration introduces a constant of integration that is determined by the initial conditions.

Kinematic Equations and Initial Conditions

Kinematic equations describe the motion of an object by relating displacement, velocity, acceleration, and time. When initial conditions such as starting acceleration, velocity, and position are provided, these equations can be tailored to describe the specific motion of the system. This approach is extended to scenarios with constant jerk, modifying the usual constant-acceleration formulas.

Energy-like Relationships in Kinematics

The relation a_x^2 = a_xi^2 + 2J(v_x - v_xi) is analogous to the energy equation in mechanics which relates velocity and displacement under constant acceleration conditions. Here, integrating the acceleration function appropriately leads to an equation linking the square of the acceleration with the difference in velocities, thereby encapsulating the cumulative effect of jerk over the motion.

Transcript

00:01 All right, here we're told that the jerk is defined as the time rate of change of the acceleration is constant.

00:10 So we're told that this quantity is a constant.

00:18 And we want to compute the acceleration as a function on time, the velocity, and the position.

00:24 From this initial equation, which is a differential equation, we know that the acceleration then is going to be equal to the jerk times the time plus the initial acceleration.

00:37 So this is the acceleration as a function of time.

00:46 Then this is answering.

00:48 The velocity is equal to the differential, the derivative of the acceleration.

00:58 No, sorry, my apologies.

01:07 The acceleration, remember that it is the derivative of the velocity as a function of time.

01:16 So if we integrate using calculus, this equation on the right side, we get the velocity is equal to the initial velocity plus one half the jerk times square times square plus the initial acceleration times the time and if since we know that the velocity is the derivative of the position as a function of time which is what i wrote up above and we're going to grade this as a function of time we integrate this with respect to time we get at the position is the initial position plus the initial velocity times the time plus the jerk over 6 to cube plus the initial acceleration t squared over 2 and now we want to compute this the the changing acceleration times the changing velocity times the jerk.

02:47 We want to prove that...

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