A 35.0 g projectile is launched by the expansion of hot gas in an arrangement sh

Question

A 35.0 g projectile is launched by the expansion of hot gas in an arrangement shown in Figure (a). The cross-sectional area of the launch tube is 1.0 cm2, and the length that the projectile travels down the tube after starting from rest is 32 cm. As the gas expands, the pressure varies as shown in Figure (b). The values for the initial pressure and volume are Pi = 1.1 106 Pa and Vi = 8.0 cm3 while the final values are Pf = 1.1 105 Pa and Vf = 40.0 cm3. Friction between the projectile and the launch tube is negligible.
(a) If the projectile is launched into a vacuum, what is the speed of the projectile as it leaves the launch tube?
m/s

(b) If instead the projectile is launched into air at a pressure of 1.1 105 Pa, what fraction of the work done by the expanding gas in the tube is spent by the projectile pushing air out of the way as it proceeds down the tube?
%

Explanation / Answer

a)

Work done by the gas W=traingle area +Rectangle area

W=0.5(Pi-Pf)(Vf-Vi)+Pf(Vf-Vi)

W=0.5(Pi+Pf)(Vf-Vi)

W=0.5*(1.3*10^6+1*10^5)(40-8)*10^-6

W=22.4 J

W=0.5mV^2

22.4 =0.5*40*10^-3*V^2

V=33.47 m/s

b)

Retarding force exerted by air

F=Pair *A=1*10^5*1*10^-4 =10N

Energy to overcome retarding force is

W(spent)=F*d =0.32*10 =3.2J

W(spent)/W =3.2/22.4 =0.1429

W(spent)/W =14.29%

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