Velocity calculation of an accelerated mass based on an increasing force

[SpaceThoughts]

A known force is doubling (egal) over a known distance, accelerating a mass.
How do I calculate the final velocity of the mass at the end of the known distance , when the mass has doubled? I don't know the time.
The mass is accelerated from 0 meter and from 0 velocity.

 

[Dale]

Since you don’t know the time, the easiest approach will be to use energy. Write down $F(x)$ then calculate $\int F(x) \ dx$ over the known distance. Assuming that is the only force then that quantity is equal to the change in KE, so you can calculate velocity.

 

[jbriggs444]

Summary:: A known force is doubling (egal) over a known distance, accelerating a mass.
How do I calculate the velocity of the mass?

A known force is doubling (egal) over a known distance, accelerating a mass.
How do I calculate the final velocity of the mass at the end of the known distance , when the mass has doubled? I don't know the time.
The mass is accelerated from 0 meter and from 0 velocity.

The first step should be to clarify the problem statement and try to assign variable names for the various parameters of the problem.

At one point you say that the force is doubling over a known distance. At another point you say that the mass has doubled. Which is it? Since masses do not usually change, I will assume that it is the force that doubles.

So you have this object with mass m at rest at the left end of a frictionless track. The track has length d. The object is subject to a variable rightward force. That force varies along the length of the track. Let us denote the rightward force experienced at position x along the track as f(x). f(0) is the force experienced at the left end. f(d) is the force experienced at the right end.

We assume that the force varies linearly (because you have not told us that it is exponential). We are told that f(d) = 2f(0).

Can you find a formula for f(x)? If you like, use $F$ for the rightward force experienced at the beginning of the scenario and write your formula in terms of $F$

 

[Point Conception]

Since you don’t know the time, the easiest approach will be to use energy. Write down $F(x)$ then calculate $\int F(x) \ dx$ over the known distance. Assuming that is the only force then that quantity is equal to the change in KE, so you can calculate velocity.

With variable force then is it just: ΔKE = ∫ F dx.
mass = 4 kg
between x=2 and x=3
F(x) 6x^2 + 4x
∫F(x) dx = 2x^3 +2x^2
ΔKE = 48 J
v = 4.89m/s