9.9 = v^2/r
9.9 * 100 = v^2
v = 99.5 m/s
Time for revolution = 2 pi r/v
= 2 pi * 100/99.5 = 6.31 seconds = .105 minutes
1/.105 = 9.5 revs/minute
frequency equals wavelength/velocity right? So how do I solve this question?
acceleration= angularvelocity^2*radius
where angular velocity is in units of radians/sec.
Solve for angular velocity above, then convert to revs/min
Note ( one rev= 2PI radian, and 1Min=50 sec.
A revolutionary cannon, with a mass of 2000kg, fires a 20kg ball horizontally. The cannonball has a speed of 140m/s after it has left the barrel. The cannon carriage is on a flat platform and is free to roll horizontally. What is the speed of the cannon immediately after it was fired? Answer in units of m/s.
Massbullet*velcity bullet= masscannon*veloicty cannon
solve for velocity of the cannon
9.9 * 100 = v^2
v = 99.5 m/s
Time for revolution = 2 pi r/v
= 2 pi * 100/99.5 = 6.31 seconds = .105 minutes
1/.105 = 9.5 revs/minute
1. First, let's calculate the linear acceleration required to produce the artificial gravity at the outer rim:
a = 9.9 m/s^2
2. We can use the formula for linear acceleration in rotational motion:
a = ω^2 * r
where ω is the angular velocity in radians/second, and r is the radius of the wheel.
3. Substituting the given values into the equation:
9.9 m/s^2 = ω^2 * 100 m
4. Solving for ω^2:
ω^2 = 9.9 m/s^2 / 100 m
ω^2 = 0.099 rad^2/s^2
5. Taking the square root of both sides to find ω:
ω ≈ √0.099 rad/s
6. To convert ω to rev/min, we need to convert it to radians/minute first:
1 rev = 2π radians
1 min = 60 seconds
ω in radians/minute = ω in radians/second * 60 seconds/min
ω in radians/minute ≈ √0.099 rad/s * 60 s/min
ω in radians/minute ≈ 3.759 rad/min
7. Finally, to convert ω to rev/min:
ω in rev/min = ω in radians/minute / (2π radians/rev)
ω in rev/min ≈ 3.759 rad/min / (2π rad/rev)
ω in rev/min ≈ 0.598 rev/min
Therefore, the frequency of the rotational motion for the wheel to produce an artificial gravity of 9.9 m/s^2 is approximately 0.598 rev/min.
Now, let's solve the second question:
1. Using the principle of conservation of momentum:
mass_bullet * velocity_bullet = mass_cannon * velocity_cannon
2. Substituting the given values into the equation:
20 kg * 140 m/s = 2000 kg * velocity_cannon
2800 kg·m/s = 2000 kg * velocity_cannon
3. Solving for velocity_cannon:
velocity_cannon = 2800 kg·m/s / 2000 kg
velocity_cannon = 1.4 m/s
Therefore, the speed of the cannon immediately after it was fired is 1.4 m/s.
acceleration = angular velocity^2 * radius
In this case, the desired acceleration is 9.9 m/s^2, and the radius of the space station wheel is 200m. Rearranging the formula to solve for angular velocity, we get:
angular velocity = sqrt(acceleration / radius)
= sqrt(9.9 / 200)
= 0.222 radians/second
Now, to convert angular velocity to revolutions per minute, we'll use the conversion factor that 1 revolution is equal to 2π radians. We also know that 1 minute is equal to 60 seconds. So:
angular velocity in revolutions per minute = (0.222 radians/second) * (1 revolution / 2π radians) * (60 seconds / 1 minute)
= 2.116 revolutions/minute
Therefore, the frequency of the rotational motion for the wheel to produce the desired "artificial gravity" effect is approximately 2.116 revolutions per minute.
Moving on to the second question about the cannon, we'll use the principle of conservation of momentum:
mass of the bullet * velocity of the bullet = mass of the cannon * velocity of the cannon
Given that the mass of the bullet is 20kg, the mass of the cannon is 2000kg, and the velocity of the bullet after leaving the barrel is 140m/s, we can calculate the velocity of the cannon:
velocity of the cannon = (mass of the bullet * velocity of the bullet) / mass of the cannon
= (20kg * 140m/s) / 2000kg
= 1.4m/s
Thus, the speed of the cannon immediately after it was fired is 1.4m/s.