Linear Momentum — Question 2

Question 2

A momentum curve encodes force and reversal

A constant-mass particle has $m=2.0\,\mathrm{kg}$ and $p_x(t)=6-4t$ in SI units for $0\le t\le3.0\,\mathrm s$.

1. Find the force and the time when its direction reverses.

2. Find displacement and distance traveled during the interval.

Question 2 - Solution

A momentum curve encodes force and reversal

(a) Derivation and calculation

Differentiate momentum and divide by constant mass: $$\boxed{F_x=\frac{dp_x}{dt}=-4.0\ \mathrm N},\qquad v_x=\frac{6-4t}{2}=3-2t.$$ $$v_x=0\Rightarrow\boxed{t_r=1.50\ \mathrm s}.$$

(b) Result and interpretation

Signed displacement and unsigned travel require different integrands: $$\boxed{\Delta x=\int_0^3(3-2t)\,dt=[3t-t^2]_0^3=0}.$$ $$\boxed{s=\int_0^3|3-2t|\,dt=2\left[\tfrac12(1.50)(3)\right]=4.50\ \mathrm m}.$$ The second half retraces the first half in the opposite direction.

Physical check

The momentum ends at $-6$ kg m/s, equal in magnitude and opposite to its initial value.