Suppose that at instant $0$ the traffic lights turn green and the car passes through the first light at instant $t_0 \ge 0$ (time is measured in seconds). The traffic lights will stay green between the time instants $150k + 90$ and $150(k+1)$, for each integer $k$. The car will pass through the lights at the instants $t_0 + \frac{1500}{v} r$, for each nonnegative integer $r$.

Therefore, a necessary and sufficient condition for a non-stopping trip is that, for every integer $r$, $\frac{t_0}{150} + \frac{10}{v}$ equals an integer plus a number between $0$ and $\frac{3}{5}$. This is clearly possible if $\frac{10}{v}$ is an integer (with any $t_0$ between $0$ and $90$) or if $\frac{10}{v}$ is half of an odd integer (with any $t_0$ between $0$ and $15$).

Let's show that these are the only possible situations. If $\frac{10}{v}$ is irrational then by Kronecker theorem the fractional part of $\frac{t_0}{150} + \frac{10}{v}$ is dense in $(0, 1)$ and thus admits values between $\frac{3}{5}$ and $1$. If $\frac{10}{v}$, say, $\frac{10}{v} = \frac{p}{q}$, $\gcd(p, q) = 1$, then the fractional part of $\frac{10}{v}$ is $\frac{pr}{q}$ takes all values of the form $\frac{s}{q}$, $0 \le s \le q-1$ (for instance, consider $r = s \cdot p^{-1} \mod q$). So consecutive values of the fractional part of $\frac{t_0}{150} + \frac{10}{v}$ are $\frac{1}{q}$ apart; the fractional part of $\frac{t_0}{150} + \frac{10}{v}$ takes values between $\frac{3}{5}$ and $1$ if $\frac{1}{r} \le 1 - \frac{3}{5} \iff r \ge 3$. Thus $r \le 2$, that is, $\frac{10}{v}$ is an integer or half of an odd integer.

We conclude that the possible values for the speed of the car are $\frac{20}{k}$ m/s, for every positive integer $k$.