A tractor pulls a trailer at a constant v_1 = 15 km/h on a…

A tractor pulls a trailer at a constant $v_1 = 15\ \mathrm{km/h}$ on a road with friction coefficient $\mu_1 = 0.01$. It then drives onto a road with $\mu_2 = 0.1$. What is its speed there if its power output stays the same?

Explanation of Method

Tractor speed on a rougher road at the same power.

Step 1: Constant power

At constant speed the pulling force equals the friction, so $P = \mu mg\,v$. With the same $P$ and $m$: \[\mu_1 v_1 = \mu_2 v_2 \;\Longrightarrow\; v_2 = v_1\frac{\mu_1}{\mu_2} = 15 \cdot 0.1.\]

\[v_2 = 1.5\ \mathrm{km/h}\]

$F = \mu mg$ $v\ (\mathrm{km/h})$ $\mu_1: \ 15\ \mathrm{km/h}$ $\mu_2 = 10\mu_1: \ 1.5\ \mathrm{km/h}$ $P = Fv = \text{same area}$

With constant power the product $Fv$ (the rectangle's area) is fixed. Ten times the friction force means one tenth of the speed.

Result

\[\boxed{\,v_2 = 1.5\ \mathrm{km/h}\,}\]

Remark

Check. The original result is correct (assuming friction is the only resistance and the engine can deliver the same power at the lower speed).