A 37.0-g projectile is launched by the expansion of hot gas in an arrangement sh

Question

A 37.0-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 cm^2, 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 P_i = 1.0 times 10^6 Pa and V_i = 8.0 cm^3 while the final values are P_f = 1.0 times 10^5 Pa and V_f = 40.0 cm^3. Friction between the projectile and the launch tube is negligible. If the projectile is launched into a vacuum, what is the speed of the projectile as it leaves the launch tube? m/s If instead the projectile is launched into air at a pressure of 1.0 times 10^5 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

b)

As we know that s

   pressure P= force / area

Force exeted by air in front of projectile

Fair = PA

       = (1.0 x 105Pa) (10^-4 m2)       

       = 10 N

Now Work= FdS (Force x displacement)

   so the energy spent will be

   Wspent = Fr s

        = Fr(0.32 m)

        = 32 Joule

   the fraction will be

   Wspent / W

Where W = area of triangle + Rectangle area

= 1/2 (Pi - Pf)(Vf-Vi) - Pf(Vf-Vi)

= 1/2 (Pi + Pf)(Vf-Vi)

= 0.5 *( 1x106 - 1 x 105)( (40x10-6 - 8x10-6)

= 22.4 Joules

Now Wspent / W = 3.2/22.4 =0.1428

=14.29 % Ans

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