A 40.0g projectile is launched by the expansion of hot gas in anarrangement show

Question

A 40.0g projectile is launched by the expansion of hot gas in anarrangement shown. The cross-sectional area of the launchtube is 1.0 cm^2, and the length that the projectile travels downthe tube after starting from rest is 32 cm. As the gas expands, thepressure varies. The values for the initial pressure andvolume are Pi = 1.0 x 10^5 Pa and Vi = 8.0 cm^3 while the finalvalues are Pf = 11 x 10^5 Pa and
Vf = 40.0 cm^3. Friction between the projectile and thelaunch tube is negligible.

(a) If the projectile is launched into a vacuum, what is the spedof the projectile as it leaves the launch tube?

(b) If instead the projectile is launched into air at a pressure of1.0 x 10^5 Pa, what fraction of the work done by the expanding gasin the tube is spent by the projectile pushing air out of the wayas it proceeds down the tube?

Explanation / Answer

(a)    to find the speed first we find the workdoneby the gas on the projectile    the workdone is the area under the curve inthe PV diagram    Wby gas = (triangular area)+(rectangular area)                 = (1 / 2) (Po - Pf) (Vf -Vo) + Pf (Vf -Vo)                = (1 / 2) (Po + Pf) (Vf- Vo)                 = ......... J    according to the work-kinetic energy theoremwe get    W = KE         = (1 / 2) mv2 - 0    then the speed of the projectile willbe    v = (2 Wby gas /m)         = .......... m / s (b)    we know that    and work as    W=  (force)(displacement)    pressure = force / area    so the air in front of the projectile wouldexert a retarding force given by    Fr = PairA            = (1.0 x 105Pa) [(1.0 cm2) (1 m2 / 104cm2)]        = ........ N    so the energy spent will be    Wspent = Fr s         = Fr(0.32 m)         = ........ J    the fraction will be    Wspent / W = ........

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