Motion with variable acceleration expression in terms of t help needed !

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Alexandramartis
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Hello, I have found the questions below in a textbook and would really appreciate some help in solving them. I have attempted to answer both parts but do not really understand how to comprehensively answer the questions, so I believe that my answers must be wrong. if anyone could be of help i would enormously appreciate it 👍😊

A particle of mass 200g moves with a velocity of 2t-t^2ms^-1, at t seconds from when it is at rest at the origin.
I do understand that if you know s as a function of time, one can find the velocity by differentiation i.e. v=ds/dt
And that you can obtain the acceleration by differentiating velocity with respect of time ie. a=dv/dt=d^2s/dt^2

1.Find an expression in terms of t for the force acting on the particle.

I really do not know where to start, would I use Newton's second law
F=ma
and since it has been established that m=200 find a in terms of;

a=dv/dt
a=2-t
F=200*(2-t)
F=400-200t

Or a=v-u/t
a= 2t-t^2/t
a=2-t
F=200*(2-t)=400-200t

2. Find the time when the particle next passes through the origin

I honestly do not know where to begin, would this be where t=d^2s/dt^2 is of use?

Or since the particle is at rest at the origin does this mean;
v(t)=2t-t^2=0
Solving the quadratic 2t+t^2+0=0
t=2 and t=0

So the particle would pass through the origin again at t=2s, or is this calculating when the particle comes to rest? Sorry I am really confused
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Pangol
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(Original post by Alexandramartis)
Hello, I have found the questions below in a textbook and would really appreciate some help in solving them. I have attempted to answer both parts but do not really understand how to comprehensively answer the questions, so I believe that my answers must be wrong. if anyone could be of help i would enormously appreciate it 👍😊

A particle of mass 200g moves with a velocity of 2t-t^2ms^-1, at t seconds from when it is at rest at the origin.
I do understand that if you know s as a function of time, one can find the velocity by differentiation i.e. v=ds/dt
And that you can obtain the acceleration by differentiating velocity with respect of time ie. a=dv/dt=d^2s/dt^2

1.Find an expression in terms of t for the force acting on the particle.

I really do not know where to start, would I use Newton's second law
F=ma
and since it has been established that m=200 find a in terms of;

a=dv/dt
a=2-t
F=200*(2-t)
F=400-200t

Or a=v-u/t
a= 2t-t^2/t
a=2-t
F=200*(2-t)=400-200t

2. Find the time when the particle next passes through the origin

I honestly do not know where to begin, would this be where t=d^2s/dt^2 is of use?

Or since the particle is at rest at the origin does this mean;
v(t)=2t-t^2=0
Solving the quadratic 2t+t^2+0=0
t=2 and t=0

So the particle would pass through the origin again at t=2s, or is this calculating when the particle comes to rest? Sorry I am really confused
For part (1), you do need to use F = ma and differentiation. You can't use a = (v-u)/t becasue this equation needs you to have constant acceleration, which you don't have. (You've also made a small slip with your differentiation.)

For part (2), yes, you have found the times when the particle is at rest. Of course you have, becasue you have solved the equaton v = 0. But you don't know that the particle will be at rest the next time it is at the origin, so this doesn't help. But you do know that if the particle is at the origin, it's displacement from the origin is zero, so that is the way to go.
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Alexandramartis
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(Original post by Pangol)
For part (1), you do need to use F = ma and differentiation. You can't use a = (v-u)/t becasue this equation needs you to have constant acceleration, which you don't have. (You've also made a small slip with your differentiation.)

For part (2), yes, you have found the times when the particle is at rest. Of course you have, becasue you have solved the equaton v = 0. But you don't know that the particle will be at rest the next time it is at the origin, so this doesn't help. But you do know that if the particle is at the origin, it's displacement from the origin is zero, so that is the way to go.
Thank you very much for your reply.

1) Oh yes I see my mistake,
a=2-2t
So would F=200*(2-2t)
F=400-400t ?

2) Finding when the displacemnt from the origin is zero, would I integrate:
s= ∫ vdt
s= ∫ 2t-t^2
s= 2t^2/2 - t^3/3
s=t^2-t^3/3 + c
When s= 0
0=t^2-t^3/3 + c

I do not know what to do next, would this be right? 😊
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Pangol
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(Original post by Alexandramartis)
Thank you very much for your reply.

1) Oh yes I see my mistake,
a=2-2t
So would F=200*(2-2t)
F=400-400t ?

2) Finding when the displacemnt from the origin is zero, would I integrate:
s= ∫ vdt
s= ∫ 2t-t^2
s= 2t^2/2 - t^3/3
s=t^2-t^3/3 + c
When s= 0
0=t^2-t^3/3 + c

I do not know what to do next, would this be right? 😊
(1) Yes, that's F = ma with your a from differentiation.

(2) Right idea, but you can't get anywhere if you don't know what c is. You need to use the fact that you are measuring displacement from the origin, and you know where the particle is when t = 0. Use this to find c, and then it will be easy to finish off.
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Alexandramartis
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(Original post by Pangol)
(1) Yes, that's F = ma with your a from differentiation.

(2) Right idea, but you can't get anywhere if you don't know what c is. You need to use the fact that you are measuring displacement from the origin, and you know where the particle is when t = 0. Use this to find c, and then it will be easy to finish off.
Thank you for your reply ✌️;

1) So is F=400-400t wrong? Sorry I am not sure how to fix this.

2) Yes, that is what stopped me progressing any further. I am really sorry but I am confused, you say I know where the particle is when t=0, do you mean the origin? I am not sure what do from here !
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Pangol
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(Original post by Alexandramartis)
Thank you for your reply ✌️;

1) So is F=400-400t wrong? Sorry I am not sure how to fix this.

2) Yes, that is what stopped me progressing any further. I am really sorry but I am confused, you say I know where the particle is when t=0, do you mean the origin? I am not sure what do from here !
You've got all the right ideas in (1). The only error you have made is that the mass is not 200 kg.

In (2), yes, as they tell you, the particle is at the origin when t = 0. So what is the value of s when t = 0? This is how you can find c.
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Alexandramartis
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(Original post by Pangol)
You've got all the right ideas in (1). The only error you have made is that the mass is not 200 kg.

In (2), yes, as they tell you, the particle is at the origin when t = 0. So what is the value of s when t = 0? This is how you can find c.
1) Oh silly me 200g = 0.2 kg
F=0.2(2-t^2)
F=0.4-0.4t

2) Sorry I am still stuck 😳
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Pangol
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(Original post by Alexandramartis)
2) Sorry I am still stuck 😳
You have got as far as s = t2 - t3/3 + c.

You know that initially, the particle is at the origin. This means that when t = 0, s = 0.

Put t = 0 and s = 0 into your equation to find c.
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Alexandramartis
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(Original post by Pangol)
You have got as far as s = t2 - t3/3 + c.

You know that initially, the particle is at the origin. This means that when t = 0, s = 0.

Put t = 0 and s = 0 into your equation to find c.
2) Ok so,
s=t^2-t^3/3 + c
0=t^2-t^3/3 + c
C=t^3/3-t^2 ?

Or 0=0^2-0^3/3 + c ?
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Pangol
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(Original post by Alexandramartis)
2) Ok so,
s=t^2-t^3/3 + c
0=t^2-t^3/3 + c
C=t^3/3-t^2 ?

Or 0=0^2-0^3/3 + c ?
I don't know how much clearer I can put it than "Put t = 0 and s = 0 into your equation to find c".

The displacement of the particle is zero when the time is zero. Of course you have to put them both equal to zero.
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Alexandramartis
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(Original post by Pangol)
I don't know how much clearer I can put it than "Put t = 0 and s = 0 into your equation to find c".

The displacement of the particle is zero when the time is zero. Of course you have to put them both equal to zero.
Into v=2t-t^2 ? Or into s=t^2-t^3/3 + c sorry I am just confused
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Pangol
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(Original post by Alexandramartis)
Into v=2t-t^2 ? Or into s=t^2-t^3/3 + c sorry I am just confused
This whole part of the conversation is about how you can find a complete equation for the displacement of the particle at any time, so that you can work out the times when the paticle is back at the origin (that is, when its displacement is zero).

You have got as far as s = t2 - t3/3 + c. You need to know that the constant c is so that you have an equation you can use.

You know that initially, the particle is at the origin. That is, s = 0 when t = 0. So put these values into your equation to find out what c is.

Velocity has nothing to do with it. And how can you put s = 0 in the velocity equation anyway?
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Alexandramartis
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s=t^2-t^3/3 + c
0= 0^2-0^3/3 + c
c=0

So is that correct that c=0?
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Pangol
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(Original post by Alexandramartis)
s=t^2-t^3/3 + c
0= 0^2-0^3/3 + c
c=0

So is that correct that c=0?
You should be able to see that it must be correct. If c was anything other than zero, would your equation say that it is at the origin when t = 0?
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Alexandramartis
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(Original post by Pangol)
You should be able to see that it must be correct. If c was anything other than zero, would your equation say that it is at the origin when t = 0?
Oh ok 😁

How do I progress from here though?
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Pangol
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(Original post by Alexandramartis)
Oh ok 😁

How do I progress from here though?
Well, why did you want to find an expression for the displacement at any time in the first place? What is it that you are trying to find?
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Alexandramartis
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(Original post by Pangol)
Well, why did you want to find an expression for the displacement at any time in the first place? What is it that you are trying to find?
I am trying to find the time when the particle next passes through the origin, so I am measuring the displacement from the origin. 😊
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Pangol
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(Original post by Alexandramartis)
I am trying to find the time when the particle next passes through the origin, so I am measuring the displacement from the origin. 😊
Yes indeed. And now you have an equation that tells you the displacement from the origin at any time. This should make it easy to find the times when the particle is at the origin.
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Alexandramartis
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(Original post by Pangol)
Yes indeed. And now you have an equation that tells you the displacement from the origin at any time. This should make it easy to find the times when the particle is at the origin.
But how do I know when to find when the particle is at the origin, is this when s=0? I really am confused.
if s=t^2-t^3/3 + 0
0=t^2-t^3/3 + c
0=t^2-t^3/3
0=t^2(1-t/3)
t=0s
And;
1-t/3=0
-t/3=-1
-t=-3
t=3 s

So the time when the particle next passes through the origin would be t=3s?
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Pangol
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(Original post by Alexandramartis)
But how do I know when to find when the particle is at the origin, is this when s=0? I really am confused.
Yes - the equation you have for s tells you the displacement of the particle from the origin. If the particle is at the origin, then it's displacement from the origin is obviously zero. So you need to solve s = 0, as you have done.
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