Problem — Two cars overtaking

Problem

Summary: two coupled motions. Two cars start from the same line at the same instant. The first starts from rest with $a = 2\;\text{m/s}^2$; the second is already moving at constant velocity $v = 12\;\text{m/s}$. When and where does the first overtake the second?

Solution

Equations of motion. We choose the origin at the starting line. The first car undergoes uniformly accelerated motion from rest, the second moves at uniform velocity:

$x_1(t) = \frac{1}{2}\,a\,t^2 = t^2, \qquad x_2(t) = v\,t = 12\,t$

Overtaking condition. Overtaking occurs when the two positions coincide, $x_1 = x_2$:

$t^2 = 12\,t \;\Rightarrow\; t\,(t - 12) = 0$

The solution $t = 0$ is the common start; the actual overtaking is the other one:

$\ev{t = 12\;\mathrm{s}}$

Overtaking position. Substituting into either equation:

$x = 12\,t = 12 \cdot 12 = 144\;\text{m}$

$\ev{x = 144\;\mathrm{m}}$

Up to $t = 12\;\text{s}$ the second car is ahead; from that point on the first, which keeps accelerating, overtakes it for good.

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Topics: Cinematica Concepts: Moto uniformemente accelerato · Velocità media