I'm not sure how to address any of these. I"ve attempted them but each time I do, I've gotten different answers. I'm not even sure where to begin.
1. The Duke’s efforts to conquer the Count are starting to really falter. He has been forced back away from the walls quite a bit because his weaponry isn’t as effective. He decides to unleash his most powerful weapon, a huge catapult (double the range, in theory, of normal catapults) and fire it at the Count’s castle. The catapult is fired from 1.000km away at the same exact altitude as the count’s wall. The stone (mass of 1576.3 kg) is perfectly round and has a diameter of 1.500m. The duke carefully calculates the angle of fire for his catapult and fires. If he used an
angle of 35 degrees (which according to his calculations, should hit) he comes up short. How short does he end up (hint: figure out why he came up short)? (The drag coefficient can be assumed to be 0.5)
vt=sqrt(2mg/DpA)= sqrt(2(1576.3)(9.81)/.5(1.276)(1.767)= 165.586
vt=mg/b; b=mg/vt; 93.387
165.586/y=cos(theata); 165.586/cos(35)=
202.14.
What would be the units on this?
4. The duke used weak hinges to fasten the doors of the gate to the walls of the camp. If a force of 25N will dislodge each hinge (there are 4 of them) and the device (essentially a pendulum) is dropped 3.45m (from highest point when it is pulled back to the point it hits the door), how fast is the door moving when it breaks free?
I don't know how to even begin this.
To find the answer to the first question, let's break it down step by step:
1. Calculate the initial velocity of the catapult stone using the given formula: vt = sqrt(2mg / (d * p * A)), where
- vt is the initial velocity
- m is the mass of the stone (1576.3 kg)
- g is the acceleration due to gravity (9.81 m/s^2)
- d is the drag coefficient (0.5)
- p is the air density (you haven't provided this value, so let's assume it to be 1.276 kg/m^3)
- A is the cross-sectional area of the stone (π * (diameter/2)^2 = π * (1.5/2)^2 = 1.767 m^2)
Substituting the values into the formula: vt = sqrt(2 * 1576.3 * 9.81 / (0.5 * 1.276 * 1.767)) ≈ 165.586 m/s
2. Calculate the horizontal distance the stone travels (range) using the equation: x = vt * t * cos(theta), where
- x is the range we want to find
- theta is the launch angle (35 degrees)
- t is the time of flight, which we can calculate using the formula: t = 2 * vt * sin(theta) / g
Substituting the values into the formula: t = 2 * 165.586 * sin(35) / 9.81 ≈ 7.851 seconds
Then: x = 165.586 * 7.851 * cos(35) ≈ 202.14 meters
The units for the range (x) would be meters.
For the second question about the door speed when it breaks free, we need to use the principle of conservation of energy. Here's how you can solve it:
1. Calculate the potential energy at the highest point: PE = mgh, where
- m is the mass of the door (you haven't provided this value)
- g is the acceleration due to gravity (9.81 m/s^2)
- h is the height (3.45 m)
2. Convert the potential energy to kinetic energy at the moment the hinge breaks: KE = 1/2 * m * v^2, where
- KE is the kinetic energy
- v is the velocity we want to find
Since the potential energy is converted entirely into kinetic energy (neglecting any energy losses due to friction), we can equate the two equations and solve for v.
PE = KE
mgh = 1/2 * m * v^2
Simplifying the equation: v = sqrt(2gh)
Substituting the values: v = sqrt(2 * 9.81 * 3.45) ≈ 11.77 m/s
Therefore, the door would be moving at approximately 11.77 m/s when it breaks free.
I hope this helps you understand how to approach these types of problems! Let me know if you have any further questions.