Vector analysis question on acceleration

In summary, the conversation discusses the assertion that a moving particle's acceleration is 0 at the instant t = 3, given its maximum speed at that time. However, the group concludes that this statement is false, as the derivative of velocity does not necessarily equal 0 at t = 3 and the direction of the velocity can still be changing. The acceleration can be shown to be non-zero at t = 3 using an expression for acceleration in terms of tangential and normal components.
  • #1
jaejoon89
195
0

Homework Statement



A moving particle reaches its max. speed at the instant t = 3. (Before and after 3, its speed is less.) It follows that the particle's acceleration is 0 at the instant t = 3... Show that this is FALSE

Homework Equations



v = dR/dt
a = d^2 R / dt^2

The Attempt at a Solution



How do I show this is false? The derivative of velocity is acceleration so I would think it's true and is indeed 0. This is on a chapter for vector analysis on acceleration and curvature.
 
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  • #2
If the particle's acceleration were 0 at t=3, why would it slow down?:wink:
 
  • #3
But the acceleration has to be 0 because at this point the speed is maximum... so on one side there should be negative acceleration and on the other side positive.
 
  • #4
No, if the acceleration at t=3 were zero, then if you measured the speed of the particle a very short time later it would be unchanged.

It's true that [tex]\frac{d}{dt} ||\vec{v}||=0[/tex] at t=3, but that doesn't necessarily mean [tex]||\vec{a}||=0[/tex] at t=3.

This rests on the fact that [tex]\vec{v}[/tex] is a vector, it has both magnitude and direction and just because its magnitude isn't changing doesn't mean it's direction can't be changing. If it's direction is changing, then [tex]\vec{a}=\frac{d\vec{v}}{dt}\neq0[/tex] :wink:

There is an expression for [itex]\vec{a}[/itex] in terms of its tangential and normal components that you should know (it involves curvature), use that to prove that the acceleration is non-zero at t=3!:smile:
 

What is vector analysis?

Vector analysis is a mathematical tool used to study and analyze quantities that have both magnitude and direction. It is commonly used in physics and engineering to analyze motion, forces, and other physical phenomena.

What is acceleration?

Acceleration is the rate of change of velocity with respect to time. It is a vector quantity, meaning it has both magnitude (speed or rate of change) and direction.

How is acceleration represented in vector analysis?

In vector analysis, acceleration is represented as a vector with its magnitude and direction. The magnitude of acceleration is the rate at which the velocity is changing, while the direction of acceleration is the direction in which the velocity is changing.

What is the difference between average and instantaneous acceleration?

Average acceleration is the change in velocity over a specific period of time, while instantaneous acceleration is the acceleration at a specific moment in time. Average acceleration is calculated by dividing the change in velocity by the change in time, while instantaneous acceleration is calculated by taking the derivative of the velocity with respect to time.

How is acceleration related to other physical quantities?

Acceleration is related to other physical quantities such as velocity, displacement, and time through various equations. For example, the equation a = (vf - vi) / t can be used to calculate acceleration using the final and initial velocities and the time interval. Acceleration is also related to force through Newton's Second Law of Motion, which states that force is equal to mass times acceleration (F = ma).

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