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 $a_{x}, v_{x}$ 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

Acceleration-Velocity Relationship

The relation that connects the square of the acceleration with the change in velocity under constant jerk conditions is an example of how integrated kinematic equations can yield relations analogous to energy methods. This emerges from integrating the differential equation and showcases the interdependence of acceleration and velocity in dynamic systems.

Initial Conditions

Initial conditions, such as initial acceleration, velocity, and position, are essential when integrating differential equations. They determine the specific values of the integration constants, ensuring that the derived equations accurately reflect the system's state at the start of the motion.

Constant Jerk

Jerk is the third derivative of position with respect to time, representing the rate of change of acceleration. In physics and engineering, assuming constant jerk allows one to model motion more accurately in scenarios where acceleration itself changes at a constant rate. This concept extends the basic constant acceleration model to include higher-order effects in motion analysis.

Integration in Kinematics

The process of integrating a constant jerk to derive acceleration, velocity, and position equations is fundamental in kinematics. Each integration introduces a constant of integration that is later determined using initial conditions. This stepwise integration is a key method in solving differential equations related to motion.

Transcript

00:01 Okay, we're looking at question 2 .33, and we're looking at something called the jerk, which is the time rate of change of acceleration.

00:13 Let's write that.

00:15 J equals d -a -d -t.

00:20 We'll just write it like that, we won't write it in vector form.

00:25 Okay, so we want to determine, or we're assuming that the object is moving in one dimension and the jerk is constant.

00:39 We want to find expressions for acceleration, velocity, and position, given that the initial acceleration, velocity, and position are as shown there.

00:51 And then for b, we want to show relation.

00:55 So let's do this.

01:00 This is how the kinematics for constant acceleration are also derived in a very similar way.

01:07 So what we're going to start with here is we're going to take our expression j equals d a d t and multiply the d t and let's put we'll put the d a on the left hand side so d a equals j d t we integrate both sides we integrate a from some initial acceleration to some final acceleration and we integrate the time from zero to some time t okay, so on the left -hand side we get a final minus a initial.

01:52 I think they may say a x initial, but i'm just going to say a initial.

01:57 It's basically the same thing, right? so a final minus a initial equals j times t.

02:06 So a final equals a initial plus j times t.

02:13 Okay, there is your first one.

02:16 Straightforward.

02:17 Now let's do the velocity equation.

02:22 So for this we say dv d t equals a.

02:28 So the integral of dv equals the integral of a d t.

02:35 The right hand side we integrate from zero to t.

02:39 The left hand side we integrate from some initial velocity to some final velocity.

02:44 So we get v final minus v initial equals and now we need to put an expression in here.

02:53 What we just solve for, a final, it might be better to write it, well, we can leave it as a final, but we just need to realize that a final is that same a we have in our integral, right? so the a final, we could also write as a of, a of t.

03:18 So we have, let's pull that in, a initial, plus jt, dt...

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