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? 39.45 Your response differs from the correct answer by more than 10%. Double check your calculations, 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 the tube is spent by the projectile pushing air out of the way as it proceeds down the tube? 11.11 Your response differs from the correct answer by more than 10%. Double check your calculations.%
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To solve this problem, we can use the principles of work and energy conservation.
First, let's calculate the work done by the gas on the projectile as it moves down the tube. The work done can be determined using the formula:
W = ΔKE + ΔPE
Where W is the work done, ΔKE is the change in kinetic energy, and ΔPE is the change in potential energy.
Since the projectile starts from rest, its initial kinetic energy is zero (ΔKE = 0).
The change in potential energy can also be neglected since the launch tube is horizontal, and there is no change in height for the projectile (ΔPE = 0).
Therefore, the work done by the gas on the projectile is simply the change in kinetic energy.
Now, the change in kinetic energy can be calculated using the equation:
ΔKE = KE_f - KE_i
Where KE_f is the final kinetic energy of the projectile and KE_i is the initial kinetic energy (zero in this case).
Next, we need to find the final kinetic energy of the projectile. We can use the equation:
KE = (1/2)mv^2
Where KE is the kinetic energy, m is the mass of the projectile, and v is the speed of the projectile.
Given that the mass of the projectile is 37.0 grams (or 0.037 kg), we can substitute these values into the equation to calculate the final kinetic energy.
Now, the work done by the gas on the projectile is equal to the change in kinetic energy, so we can determine the speed of the projectile. We can rearrange the equation:
ΔKE = KE_f - KE_i
To solve for v:
KE_f = ΔKE + KE_i
v = √((2 * KE_f)/m)
By substituting the calculated values for the final kinetic energy and mass of the projectile, we can find the speed of the projectile.
After calculating the speed of the projectile launched into a vacuum (39.45 m/s), please check your calculations to see if there is any error in your work.
To determine the fraction of the work done by the expanding gas used to push air out of the way when the projectile is launched into air, we need to find the work done by the gas on the projectile plus the work done by the gas to push the air out.
The work done to push the air out can be calculated by taking the difference in volume and multiplying it by the pressure.
Now, we can find the fraction of the work done by the expanding gas spent by the projectile pushing air out of the way as it proceeds down the tube. This can be determined by dividing the work done by the gas on the projectile by the combined work done by the gas on the projectile and the work done to push the air out.
By substituting the appropriate values into the equation, we can calculate this fraction (11.11%). Please recheck your calculations to ensure accuracy and correct any potential errors.