Work in a thermodynamic process

[RslM]

Hello, could someone please let me know if I have worked this problem out correctly, or if I need to try again? Thank you.

Homework Statement

A 40.0 g projectile is launched by the expansion of hot gas in an arrangement showing in Figure P12.4a (attached file). 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 P12.4b. The values for the initial pressure and volume are P_i = 11 x 10⁵ Pa and V_i = 8.0 cm³ while the final values are P_f = 1.0 x 10⁵ Pa and V_f = 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?

B) If instead the projectile is launched into air at a pressure of 1.0 x 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?

Homework Equations

I chose to use:
Work = the area under a curve
KE = 1/2mV²

The Attempt at a Solution

Area under the curve:

P_i = 11 x 10⁵ Pa
P_f = 1.0 x 10⁵ Pa

V_i = 8.0 cm³
V_f = 40.0 cm³

ΔP = 1.0 x 10⁵ Pa - 11 x 10⁵ Pa = -10 x 10⁵ Pa

ΔV = 40 cm³ - 8.0 cm³ = 32 cm³
= 3.2 x 10^-5 m³

Area of triangle under curve = (1/2)(-10 x 10⁵ Pa)(3.2 x 10^-5 m³) = -16 J

Area of rectangle under curve = (-1.0 x 10⁵ Pa)(3.2 x 10^-5 m³) = -32 J

W = -32 J + (-16 J) = -48 J


KE = 1/2mv²

v = √(2KE/m)

KE = 48 J
m = 0.040 kg

V = √((2(48J))/0.040 kg) = 48 m/s

B) W = (-1.0 x 10⁵ Pa)(3.2 x 10^-5 m³) = -32 J

-32 J/-48 J = 2/3

2/3 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.

 

[BvU]

No replies so far, perhaps because your question isn't quite legit within PF. PF helpers help to come closer to a solution by asking questions and giving hints. PF is not about ensuring eveyone scoring an A+ by stamp-approving their work.
That being said, you've done the work, found an answer, checked it. Is there something you have checked that you think doesn't fit ? Some small step you feel unsure about ?

 

[RslM]

Thank you for the reply.