Finding distance traveled....? Mechanics questions

[ztalira]

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?

 

[SteamKing]

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?

 

[ztalira]

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]?

 

[SteamKing]

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.

 

[ztalira]

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)?

 

[SteamKing]

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?