Elastic Collisions in One Dimension — Question 5

Question 5

A finite wall mass quantifies the fixed-wall approximation

A $1\,\mathrm{kg}$ cart moving at $10\,\mathrm{m/s}$ collides elastically with a stationary free block of mass $99\,\mathrm{kg}$. Motion is one dimensional.

1. Find both final velocities and the block's acquired energy.

2. Compare with the ideal fixed-wall result and express the incident cart's velocity error.

Question 5 - Solution

A finite wall mass quantifies the fixed-wall approximation

(a) Derivation and calculation

Apply the finite-mass elastic collision formulas: $$\boxed{v_1=\frac{1-99}{1+99}(10)=-9.8\ \mathrm{m/s}},$$ $$\boxed{v_2=\frac{2(1)}{1+99}(10)=0.20\ \mathrm{m/s}}.$$ $$\boxed{K_{2f}=\tfrac12(99)(0.20)^2=1.98\ \mathrm J}.$$

(b) Result and interpretation

The fixed-wall approximation gives $v_1=-10$ m/s. Its discrepancy is $$|v_{1,\mathrm{exact}}-v_{1,\mathrm{wall}}|=0.20\ \mathrm{m/s}=0.02u.$$ The finite target recoil carries energy and momentum; neglecting it is accurate here to 2 percent of the incident speed.

Physical check

The projectile retains 48.02 J, so the final energies sum to the initial 50 J.