Finding distance traveled....? Mechanics questions

AI Thread Summary
The discussion focuses on solving a mechanics problem involving a point mass subjected to a time-dependent force. The key challenge is determining the velocity and distance traveled by the mass, starting from rest. Participants clarify that since acceleration is not constant, the final velocity must be found by integrating the varying acceleration over time. There is confusion regarding how to set up the integral and evaluate it, particularly with respect to the boundaries for the integration. Understanding calculus and integration techniques is emphasized as essential for solving the problem effectively.
ztalira
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Homework Statement


A point mass,m, is constrained to move in one-dimension and is acted on buy a force that depends on time in the following way:
F[x](t)=ƒ[o]+αt+βt^2
where ƒo,α, and β are constants . In terms of the quantities given, answer the following:
If the object starts off at rest at t=0, find its velocity at a later time, t[f]
Find the distance the mass has traveled from t=0 to t=t[f]

Homework Equations


v=at
F=ma

The Attempt at a Solution


I believe I found the velocity at t[f] by, having
a=F/m then
v=(F/m)t
and for t[f]
v[f]=((ƒ[o]+αt[f]+βt[f]^2)/(m))*t[f]but from there, I'm kinda stuck.
I know that the distance must be the integral of the velocity, but, with velocity changing, how can I find the integral?
Or is there perhaps another way?
 
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ztalira said:

Homework Statement


A point mass,m, is constrained to move in one-dimension and is acted on buy a force that depends on time in the following way:
F[x](t)=ƒ[o]+αt+βt^2
where ƒo,α, and β are constants . In terms of the quantities given, answer the following:
If the object starts off at rest at t=0, find its velocity at a later time, t[f]
Find the distance the mass has traveled from t=0 to t=t[f]

Homework Equations


v=at
F=ma

The Attempt at a Solution


I believe I found the velocity at t[f] by, having
a=F/m then
v=(F/m)t
and for t[f]
v[f]=((ƒ[o]+αt[f]+βt[f]^2)/(m))*t[f]but from there, I'm kinda stuck.
I know that the distance must be the integral of the velocity, but, with velocity changing, how can I find the integral?
Or is there perhaps another way?
You're partially correct.
v = at only when a = constant. Is a = constant here?
 
SteamKing said:
You're partially correct.
v = at only when a = constant. Is a = constant here?

a is not a constant, meaning my answer of v[f] is wrong. I could find dv/dt for t[f], but that would only give me the acceleration at that time. How could I find, then, the final velocity at t[f]?
 
ztalira said:
a is not a constant, meaning my answer of v[f] is wrong. I could find dv/dt for t[f], but that would only give me the acceleration at that time. How could I find, then, the final velocity at t[f]?
Since a = dv / dt, you'll have to integrate acceleration to find velocity.
 
SteamKing said:
Since a = dv / dt, you'll have to integrate acceleration to find velocity.
ok. so,
Fx(0)=f[o]
Fx=ma
a=f[o]/m (let this be the first acceleration, or a[1])
and for t=t[f]
Fx(tf)=f[o]+αt[f]+βt[f]^2
Fx=ma
a=(f[o]+αt[f]+βt[f]^2)/m (let this be acceleration #2, or a[2])
Δa=a[2]-a[1]
Δa=(1/m)(αt[f]+βt[f]^2)
So, I integrate Δa with respect to t?
but, the boundaries for the integral would be (0.t[f])?
I'm confused with how to integrate this, or how to evaluate t[f] (either as a value or variable)?
 
ztalira said:
ok. so,
Fx(0)=f[o]
Fx=ma
a=f[o]/m (let this be the first acceleration, or a[1])
and for t=t[f]
Fx(tf)=f[o]+αt[f]+βt[f]^2
Fx=ma
a=(f[o]+αt[f]+βt[f]^2)/m (let this be acceleration #2, or a[2])
Δa=a[2]-a[1]
Δa=(1/m)(αt[f]+βt[f]^2)
So, I integrate Δa with respect to t?
but, the boundaries for the integral would be (0.t[f])?
I'm confused with how to integrate this, or how to evaluate t[f] (either as a value or variable)?
Have you studied calculus any? Do you know how to integrate a function of a single variable?
 

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