Work and Energy
Beginner
mechanical energy and conservation
Source: Principles of Physics
Problem
A bundle of mass $m$ starts up an incline of angle $\theta$ with an initial kinetic energy $K_0$. The coefficient of kinetic friction between the bundle and the incline is $\mu_k$.
How far, $d$, will the bundle slide up the incline before coming to rest?
[Q1] $d = \frac{K_0}{mg(\sin\theta + \mu_k \cos\theta)}$
Using the work-energy theorem, the change in kinetic energy from the start to the highest point is $\Delta K = 0 - K_0 = -K_0$. This change is caused by the negative work done by gravity, $W_g = -mg d \sin\theta$, and by kinetic friction, $W_f = -(\mu_k N)d = -(\mu_k mg \cos\theta)d$. Setting the total work equal to the change in kinetic energy:
$$-mgd(\sin\theta + \mu_k \cos\theta) = -K_0$$Solving for the distance $d$ gives the result.