# Step Force

1. Jul 19, 2008

### jesuslovesu

[solved] Step Force

Never mind, i think i figured it out
1. The problem statement, all variables and given/known data
Suppose a body of mass m is subject to a constant force $$F_1$$ that acts for a length time $$t_1$$ and then the force suddenly changes to a constant $$F_2$$

Show $$v(t) = v_0 + \frac{F_1 t_1}{m} + \frac{F_2(t-t_1 )}{m}$$
$$x(t) = x_0 + v_0 t_1 + \frac{F_1 {t_1}^2}{2m} + (v_0 + \frac{F_1 t_1}{m}(t-t_1 ) ) + \frac{F_2}{2m}(t-t_1 )^2$$

2. Relevant equations

3. The attempt at a solution

I'm having a hard time with the step force. Here's how I get v(t)

$$\int_{v_0}^v dv = \int_0^{t_1} F_1/m dt + \int_{t_1}^{t} F_2/m dt$$
$$v(t) = F_1/m t_1 + F_2/m (t - t_1) + v_0$$
Which is fine, but then when it gets to x(t) is where I am having problems:

$$\int_{x_0}^{x} dx = \int_{0}^{t_1} F_1/m t_1 dt + \int_{t_1}^{t} F_2/m (t-t_1) dt + \int_0^{t_1} v_0 dt + \int_{t_1}^t v_0 dt$$

Which yields:
$$x(t) = x_0 + v_0 t_1 + (v_0)(t-t_1) + F_1/m {t_1}^2 + \frac{F_2 (t-t_1)^2 }{2m}$$

I know that in order to have a step function you need to get rid of $$F_1$$ after time t_1 because it no longer acts on the object. So I could just add $$F_1 t_1 / m (t-t_1)$$ but I'm not really sure where the $$F_1 t_1/m (t)$$ force would come into play.

So my question is: Am I doing the analysis correctly? It seems like taking care of the step function is proving to be a difficultly.

Last edited: Jul 19, 2008