The spring constant of a toy dart gun is 4442 N=m. To the gun the spring is compressed 1.5 cm. The 8 g dart, fired straight upward, reaches a maximum height of 26 m. The acceleration of gravity is 9.81 m/s2 :
Determine the magnitude of the energy dissipated by air friction during the dart's ascent.Answer in units of J.
The stored potential enery in the spring is (1/2)kX^2 = 0.50 J
At the top of the trajectory, the potential energy will have increased by M g H = (0.008 kg)(9.81 m/s^2)(26 m) = 2.04 J
Something is wrong with the numbers you have provided. The dart could not have gone that high without violating the conservation of energy law.
Well, it seems like the dart has some serious aspirations of becoming an astronaut! I mean, reaching a height of 26 meters with just a toy dart gun? That's almost as impressive as a squirrel learning the Moonwalk!
But alas, we must remember the laws of physics. Conservation of energy tells us that the potential energy at the top of the trajectory should be equal to the initial potential energy stored in the spring plus the energy dissipated by air friction. So, instead of a height of 26 meters, let's go back to reality and find the correct answer, shall we?
Given that the potential energy stored in the spring is 0.50 J and the potential energy at the top of the trajectory is 2.04 J, we can subtract these values to find the energy dissipated by air friction.
2.04 J - 0.50 J = 1.54 J
Voila! The magnitude of the energy dissipated by air friction during the dart's ascent is 1.54 J. So, the dart might not be reaching for the stars, but it's certainly making some air molecules work up a sweat!
Based on the given information, the magnitude of the energy dissipated by air friction during the dart's ascent cannot be determined accurately. However, based on the conservation of energy law, the total energy dissipated by air friction should be equal to the difference between the initial potential energy stored in the spring and the final potential energy at the top of the trajectory.
The initial potential energy stored in the spring is given as 0.50 J.
The final potential energy at the top of the trajectory is calculated as (0.008 kg)(9.81 m/s^2)(26 m) = 2.04 J.
Therefore, the magnitude of the energy dissipated by air friction during the dart's ascent can be estimated as the difference between these two values:
Energy dissipated = Final potential energy - Initial potential energy
= 2.04 J - 0.50 J
= 1.54 J
Thus, the magnitude of the energy dissipated by air friction during the dart's ascent is approximately 1.54 J.
To solve this problem, we need to use the principle of conservation of mechanical energy, which states that the initial potential energy plus the initial kinetic energy is equal to the final potential energy plus the final kinetic energy, assuming no external forces such as air friction acting on the system.
We are given the spring constant (k) of the toy dart gun, which is 4442 N/m, and the distance the spring is compressed (X), which is 1.5 cm (or 0.015 m). From this information, we can calculate the stored potential energy in the spring.
Stored potential energy in the spring = (1/2) kX^2
= (1/2) * 4442 N/m * (0.015 m)^2
= 0.499 J (approx)
Now, let's consider the initial kinetic energy of the dart. Since we are given the mass (m) of the dart, which is 8 g (or 0.008 kg), and the acceleration due to gravity (g), which is 9.81 m/s^2, we can calculate the initial kinetic energy.
Initial kinetic energy = (1/2) m v^2
= (1/2) * 0.008 kg * v^2
= 0.004 v^2 J
where v is the initial velocity of the dart.
Next, let's consider the final potential energy of the dart at its maximum height. We are given the height (H) reached by the dart, which is 26 m. Since there is no forward/backward motion, the final kinetic energy is zero. Therefore, the final potential energy is equal to the sum of the stored potential energy in the spring and the initial kinetic energy.
Final potential energy = Stored potential energy + Initial kinetic energy
= 0.499 J + 0.004 v^2 J
Using the principle of conservation of mechanical energy, we can equate the initial potential energy plus the initial kinetic energy to the final potential energy plus the final kinetic energy.
Stored potential energy + Initial kinetic energy = Final potential energy + Final kinetic energy
0.499 + 0.004 v^2 = 2.04 + 0
0.004 v^2 = 1.541
Now, something seems to be wrong with the given numbers because the dart couldn't have reached a maximum height of 26 m using these calculations. The given numbers don't satisfy the conservation of energy law, which implies that there might be some external force acting on the dart during its ascent, like air friction.
To determine the magnitude of the energy dissipated by air friction during the dart's ascent, we would need additional information, such as the time taken to reach the maximum height or the distance covered. With this information, we could calculate the work done by the external force of air friction and determine the energy dissipated.