Work, Energy and Power: Speed-Solving Techniques (10)

Question

A 2 kg block slides down a $30°$ frictionless incline of length 5 m, then continues on a horizontal surface where the coefficient of kinetic friction is $0.2$. How far does it travel on the horizontal surface before stopping? Solve in under 60 seconds using energy conservation only — no kinematics, no force diagrams.

Solution — Step by Step

Why This Works

Energy conservation lets us skip the velocity-at-the-bottom calculation entirely. The total mechanical energy at the top equals the total work done against friction by the time the block stops. One equation, one unknown.

In any problem where the question asks “how far” or “how high” and friction is the only dissipator, write $mgh = \mu m g d$ (or $\mu m g \cos\theta \cdot d$ on inclines) and solve directly.

Alternative Method

Kinematics route: find $v$ at bottom of incline using $v^2 = 2 g h = 50$, so $v = \sqrt{50}$ m/s. Then on horizontal surface, deceleration $= \mu g = 2$ m/s². Use $v^2 = 2 a d \implies 50 = 2 \cdot 2 \cdot d \implies d = 12.5$ m. Same answer, three steps instead of one.

Common Mistake