Collision and Impulse — Question 4

Question 4

Weight matters when contact is not instantaneous

A $0.50\,\mathrm{kg}$ ball strikes a horizontal floor with vertical velocity $-4\,\mathrm{m/s}$ and rebounds at $+3\,\mathrm{m/s}$. Contact lasts $0.10\,\mathrm s$; take $g=9.8\,\mathrm{m/s^2}$.

1. Find the net impulse and the upward impulse from the floor.

2. Find the average normal force and the percentage underestimate if gravity is omitted, relative to the correct force.

Question 4 - Solution

Weight matters when contact is not instantaneous

(a) Derivation and calculation

The net impulse is the momentum change, while gravity also contributes during contact: $$\boxed{J_{\mathrm{net}}=0.50[3-(-4)]=3.50\ \mathrm{N\,s}}.$$ $$J_N-mg\Delta t=J_{\mathrm{net}}\Rightarrow \boxed{J_N=3.50+0.50(9.8)(0.10)=3.99\ \mathrm{N\,s}}.$$

(b) Result and interpretation

The normal-force average and omitted-gravity error are $$\boxed{N_{\mathrm{avg}}=3.99/0.10=39.9\ \mathrm N},\qquad N_{\mathrm{omit}}=3.50/0.10=35.0\ \mathrm N.$$ $$\boxed{\frac{N_{\mathrm{avg}}-N_{\mathrm{omit}}}{N_{\mathrm{avg}}}\,100\% =\frac{4.9}{39.9}\,100\%=12.3\%}.$$

Physical check

The floor impulse must exceed the net upward impulse because gravity acts downward throughout contact.