Acceleration in terms of position

In summary, Acceleration is the second derivative of position, and when velocity is changing in the negative direction, the acceleration is also changing in the positive direction.
  • #1
yoamocuy
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0

Homework Statement


I'm given acceleration as a=-1.5*s where s is position, and I need to derive an expression for acceleration as a function of time. I am also given an initial velocity of 20 m/s and initial position of 0 m.


Homework Equations


a=-1.5*s

Characteristic Equation
s(t)=C1*e-p*t*cos(sqrt(q)*t)+C2e-p*t*sin(sqrt(q)*t)

The Attempt at a Solution


acceleration is the 2nd derivative of position, therefore a=-1.5*s is also equal to d2s/dt2=-1.5*s

d2s/dt2+1.5*s=0

I took the laplace transform of both sides to get: s2 + 1.5=0

Solving for s I get s=i*sqrt(1.5)

plugging this into the characteristic equation I get:
s(t)=C1*e0*cos(sqrt(1.5)*t)+C2*e0*sin(sqrt(1.5)*t)

at t=0 this equation becomes:
0=C1*e0*cos(0)+C2*e0*sin(0)

therefore C1=0

so s(t)=C2*sin(sqrt(1.5)*t)

Take the derivative to get:
v(t)=sqrt(1.5)*C2*cos(sqrt(1.5)*t)

at t=0 v(t)=20 therefore
20=sqrt(1.5)*C2*1

C2=16.33

that makes s(t)=16.33*sin(sqrt(1.5*t))
and v(t)=20*cos(sqrt(1.5)*t)

take the derivative to get a(t):
a(t)=-24.5*sin(sqrt(1.5)*t)

All of this seemed ok to me until I graphed all three functions and realized that according to these equations when my particle is accelerating its velocity is slowing down, which can not be possible. Did I do this question completely wrong?
 
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  • #2
yoamocuy said:
All of this seemed ok to me until I graphed all three functions and realized that according to these equations when my particle is accelerating its velocity is slowing down, which can not be possible. Did I do this question completely wrong?
No. Acceleration and velocity are generally independent, and, in particular, they can have opposite directions. For example, if the initial velocity is vi in the positive x direction, and then the velocity changes by an amount Δv in the negative x direction, then the speed (magnitude of velocity) decreases. If this change happens in a time Δt, then the average acceleration is Δv/Δt in the negative x direction. Remember, velocity and acceleration are vectors, so they each have magnitude and direction.
 
  • #3
I understand all that, but with my functions, since they are all cos and sin functions, when my particle's velocity is traveling in the negative direction my acceleration is traveling in the positive direction. It shouldn't be like that should it? Or wait.. I should be graphing this in radian rather than degrees shouldn't I?
 
  • #4
It should be exactly like that, and you'll see why once you realize what the equation a=-1.5*s really means. If you use "x" instead of s and "F/m" instead of a, you'll get:

F/m=-1.5x
F=-1.5mx

Call 1.5m "k": F=-kx

Do you recognize that equation?
 
  • #5
Oh ok, that makes it much clearer haha. Thanks
 

1. What is acceleration in terms of position?

Acceleration in terms of position is the rate at which an object's position changes over time. It is a measure of how quickly the velocity of an object is changing.

2. How is acceleration in terms of position different from regular acceleration?

Regular acceleration refers to the change in an object's velocity, while acceleration in terms of position specifically measures the change in an object's position over time. It takes into account both the magnitude and direction of an object's velocity.

3. What is the formula for calculating acceleration in terms of position?

The formula for acceleration in terms of position is a = (vf - vi) / t, where a is acceleration, vf is final velocity, vi is initial velocity, and t is time.

4. How does acceleration in terms of position affect an object's motion?

Acceleration in terms of position can either speed up or slow down an object's motion, depending on whether the acceleration is in the same direction or opposite direction as the object's velocity. If the acceleration is in the same direction, the object will speed up. If the acceleration is in the opposite direction, the object will slow down.

5. What are some real-life examples of acceleration in terms of position?

Some examples of acceleration in terms of position include a car speeding up or slowing down, a roller coaster going up or down a hill, a person jumping off a diving board, or a ball being thrown in the air and coming back down.

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