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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.

Andrei
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Aymeric Bianco pelle
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2 Answers2

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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 $$\lim_{n\to\infty} d_n=100\frac{1}{1-0.975} = \frac{100}{0.025} = 4000$$

Nicolas FRANCOIS
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  • Wow it's that simple, thank you. So this is a series problem right ? – Aymeric Bianco pelle Oct 30 '18 at 22:32
  • You can see it as a series problem, or as a sequence limit problem, right. Andrei gives you the continuous answer (where the speed varies continuously threw time, and not every second), which, strangely, gives the same answer... – Nicolas FRANCOIS Oct 31 '18 at 01:17
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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 $$d=\int_0^\infty v(t)dt=4000{\rm m}$$

Andrei
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