Find velocity-time equation from velocity-position equation

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  • #1
MatinSAR
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Homework Statement
Find velocity-time and position-time equation from velocity-position equation(Kinematics).
Relevant Equations


The velocity-position equation of an object that is moving in a straight line is . If find average velocity for .

Answer of the book starts with :

We know that :



So => and .
=>

I can't undestand how it finds out that at t=0 we have because it might be . In this case book's answer is wrong be cause we should use ...
 
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  • #2
Your relevant equations are all for the constant acceleration case. You ought not assume that here. However, it does look like the book answer assumes that. It also assumes .
As to your question, the problem statement does not mention . It does not define , but it is reasonable to suppose that means the velocity at t=0.
The solver chose to represent t at the start, so it equals 0 by definition.

Fwiw, you can deduce acceleration is constant. Differentiating wrt x yields , and in general .
 
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  • #3
haruspex said:
Your relevant equations are all for the constant acceleration case. You ought not assume that here. However, it does look like the book answer assumes that. It also assumes .
As to your question, the problem statement does not mention . It does not define , but it is reasonable to suppose that means the velocity at t=0.
The solver chose to represent t at the start, so it equals 0 by definition.

Fwiw, you can deduce acceleration is constant. Differentiating wrt x yields , and in general .
Thank you for your time.
How can we know that ?
It might be velocity in another time ... because in general we have and this is different from .
 
  • #4
MatinSAR said:
The velocity-position equation of an object that is moving in a straight line is .
You can also see directly that this equation is of the form . Compare this with the equation of constant acceleration, starting at : .

One solution, therefore, is .

And, let's not worry about units.
 
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  • #5
PeroK said:
You can also see directly that this equation is of the form . Compare this with the equation of constant acceleration, starting at : .

One solution, therefore, is .

And, let's not worry about units.
Thank you.
But we can use these if we have . Am I right?
I mean if for we have the we cannot use .
 
  • #6
MatinSAR said:
we have the we cannot use .
That's purely notation. I would take . If you take , then you have the same solution, but you need to mess about unnecessarily with the standard equations.
 
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  • #7
PS if you draw a graph (extended to ), then it's clear. is just an arbitrary label on the graph. It's the same graph however you label the points, so you might as well have .
 
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  • #8
PeroK said:
That's purely notation. I would take . If you take , then you have the same solution, but you need to mess about unnecessarily with the standard equations.
This is what I meant. If I choose for example then I can't use . Am I right?
 
  • #9
MatinSAR said:
This is what I meant. If I choose for example then I can't use . Am I right?
The full equations have and in them. For example:
AndandThe point of my first post here was that you should recognise an equation you've seen before. That was a quick way to realise we do have a constant acceleration solution.
 
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  • #10
PeroK said:
PS if you draw a graph (extended to ), then it's clear. is just an arbitrary label on the graph. It's the same graph however you label the points, so you might as well have .
So according to this, Can I say that is an equation of a line and we can find it by any given two points like and ?
 
  • #11
MatinSAR said:
So according to this, Can I say that is an equation of a line and we can find it by any given two points like and ?
Yes, there's an important link generally between kinematic equations and geometry. The trajectory of a particle is, after all, a curve of some description in 3D space.

In a physical scenario, the starting time can be literally where the motion starts (e.g. a ball being kicked). Or, it can be some arbitrary point in the trajectory (e.g. a planetary orbit).

Constant acceleration motion maps to straight lines and parabolas. Planetary and related motion maps to conic sections (circles, ellipses and hyperbolas).

A charged particle in a constant magnetic field moves in a helix.

And, more generally motion is some smooth curve in space.
 
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  • #12
PeroK said:
Yes, there's an important link generally between kinematic equations and geometry. The trajectory of a particle is, after all, a curve of some description in 3D space.

In a physical scenario, the starting time can be literally where the motion starts (e.g. a ball being kicked). Or, it can be some arbitrary point in the trajectory (e.g. a planetary orbit).

Constant acceleration motion maps to straight lines and parabolas. Planetary and related motion maps to conic sections (circles, ellipses and hyperbolas).

A charged particle in a constant magnetic field moves in a helix.

And, more generally motion is some smooth curve in space.
Thank you for your time.
 
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