Kinetic friction removes mechanical energy from a moving object and converts that energy into thermal energy.
This calculator uses the work-energy theorem and the mechanical energy balance to determine final speed,
stopping distance, friction work, and energy dissipated as heat.
1. Work done by kinetic friction
The kinetic friction force has magnitude
\[
f_k=\mu_kN,
\]
where \(\mu_k\) is the coefficient of kinetic friction and \(N\) is the normal force.
Since kinetic friction acts opposite the direction of motion, its work over a path length \(d\) is negative:
\[
W_f=-f_kd=-\mu_kNd.
\]
The thermal energy produced is the positive quantity
\[
E_{\mathrm{th}}=-W_f=\mu_kNd.
\]
2. Horizontal surface
On a horizontal surface,
\[
N=mg.
\]
Therefore,
\[
W_f=-\mu_kmgd.
\]
Because the height does not change, \(\Delta U_g=0\). The work-energy equation becomes
\[
\frac12mv_0^2-\mu_kmgd=\frac12mv_f^2.
\]
Solving for the final speed:
\[
v_f=\sqrt{v_0^2-2\mu_kgd}.
\]
The expression inside the square root must be nonnegative. If it is negative, the object stops before traveling the
full requested distance.
3. Inclined track
On an incline at angle \(\theta\),
\[
N=mg\cos\theta.
\]
Thus the friction work is
\[
W_f=-\mu_kmg\cos\theta\,d.
\]
The height change depends on the direction of motion:
\[
\Delta h=
\begin{cases}
d\sin\theta, & \text{moving up the incline},\\
-d\sin\theta, & \text{moving down the incline}.
\end{cases}
\]
The gravitational potential energy change is
\[
\Delta U_g=mg\Delta h.
\]
4. General energy balance
The calculator uses
\[
K_i+W_{\mathrm{other}}+W_f=K_f+\Delta U_g.
\]
Equivalently,
\[
K_f=K_i+W_{\mathrm{other}}+W_f-\Delta U_g.
\]
where
\[
K_i=\frac12mv_0^2,
\qquad
K_f=\frac12mv_f^2.
\]
Positive \(W_{\mathrm{other}}\) adds energy, such as a push or engine work. Negative \(W_{\mathrm{other}}\) removes
extra energy, such as braking or drag.
5. Stopping distance
If the final kinetic energy is zero, the object has stopped. On a horizontal surface with no other work,
\[
0=\frac12mv_0^2-\mu_kmgd_{\mathrm{stop}}.
\]
So
\[
d_{\mathrm{stop}}=\frac{v_0^2}{2\mu_kg}.
\]
For an inclined track, the calculator uses the general energy-loss-per-meter form:
\[
d_{\mathrm{stop}}
=
\frac{K_i+W_{\mathrm{other}}}
{f_k+mg(\Delta h/d)}.
\]
This denominator is the energy removed per meter of path. If it is zero or negative, the object does not stop by
this mechanism because gravity or other work balances or exceeds the friction loss.
6. Worked example and important correction
Suppose a \(12\ \mathrm{kg}\) box slides on a horizontal surface with
\(\mu_k=0.25\), \(v_0=6\ \mathrm{m/s}\), and \(d=18\ \mathrm{m}\).
Step 1: initial kinetic energy.
\[
K_i=\frac12mv_0^2
=
\frac12(12)(6)^2
=
216\ \mathrm{J}.
\]
Step 2: friction work over the full entered distance.
\[
W_f=-\mu_kmgd
=
-(0.25)(12)(9.81)(18)
\approx -530\ \mathrm{J}.
\]
Since the friction work over \(18\ \mathrm{m}\) would remove more energy than the box initially has,
the box cannot travel the full \(18\ \mathrm{m}\). It stops earlier.
Step 3: stopping distance.
\[
d_{\mathrm{stop}}=\frac{v_0^2}{2\mu_kg}
=
\frac{6^2}{2(0.25)(9.81)}
\approx 7.34\ \mathrm{m}.
\]
The physically correct final speed at the requested endpoint is therefore not real; the box has already stopped.
The actual energy dissipated before stopping is \(216\ \mathrm{J}\).
Formula summary
| Concept |
Formula |
Meaning |
| Kinetic friction |
\(f_k=\mu_kN\) |
Friction force magnitude. |
| Friction work |
\(W_f=-f_kd\) |
Negative work done by kinetic friction. |
| Thermal energy |
\(E_{\mathrm{th}}=-W_f\) |
Mechanical energy converted into heat. |
| Horizontal normal force |
\(N=mg\) |
Normal force on a horizontal surface. |
| Incline normal force |
\(N=mg\cos\theta\) |
Normal force on an inclined track. |
| Energy balance |
\(K_i+W_{\mathrm{other}}+W_f=K_f+\Delta U_g\) |
General work-energy equation used by the calculator. |
| Horizontal final speed |
\(v_f=\sqrt{v_0^2-2\mu_kgd}\) |
Final speed if the object reaches the full distance. |
| Horizontal stopping distance |
\(d_{\mathrm{stop}}=\frac{v_0^2}{2\mu_kg}\) |
Distance before the object stops on a horizontal surface. |
The key idea is that kinetic friction does negative work: it removes mechanical energy and converts it into thermal energy.
