Maximum distance traveled by an object with exponentialy decaying speed

Question

An object launched at $100$ m/s sees it's speed decreasing by $2.5$% every seconds. What is the distance traveled after an infinite amount of time ?

I know the formula used to find speed at any t time but that's all.

I think this would be called non-uniform deceleration and that the speed is exponentially decaying but I'm missing the knowledge and vocabulary to go further in my research, Ty.

Answer 1

Welcome to MSE.

You can use the formula : $$1+q+q^2+\dots+q^n=\frac{1-q^{n+1}}{1-q}$$ Here, if $u_i$ is the distance traveled between instants $i$ and $i+1$ (seconds), you have $u_i=100\times 0.975^i$, so the total distance during the first $n$ seconds should be $$d_n=u_0+u_1+\dots+u_{n-1} = 100(1+0.975+0.975^2+\dots+0.975^{i-1}) = 100\frac{1-0.975^n}{1-0.975}$$ When $n$ tends to $\infty$, the limit of this expression is $$\lim_{n\to\infty} d_n=100\frac{1}{1-0.975} = \frac{100}{0.025} = 4000$$

Answer 2

The velocity is given by the differential equation $$\frac {dv}{dt}=-0.025v$$The solution for this, with the initial condition is $$v=100e^{-0.025t}{\rm m/s}$$ The distance can be written as $$d=\int_0^\infty v(t)dt=4000{\rm m}$$