A hot-air balloon has a total mass of 720 kg. It accelerates downwards (towards the ground) at 2.7m/s^2 even though an upwards lift force in acting on it.
Find
a:Find the lift force acting on the balloon
B)What must be the total mass that needs to be thrown overboard if the operator on the hot-air balloon decides to accelerate upwards at 3.6 m/s^2 ?
Assume the upwards lift force remains the same.
a. We'll call downward acceleration
(-), and upward (+).
a + g = -2.7 m/s^2,
a - 9.8 = -2.7,
a = -2.7 + 9.8 = 7.1 m/s^2.
F = ma = 720 kg * 7.1 m/s^2 = 5112 N.
b. a + g = 3.6 m/s^2,
a - 9.8 = 3.6,
a = 3.6 + 9.8 = 13.4 m/s^2.
F = ma = 5112 N,
m * 13.4 = 5112,
m = 5112 / 13.4 = 381.5 kg, max.
m = 720 - 381.5 = 338.5 kg to be thrown overboard.
net force=ma
lift-mg=ma but a=-2.7
solve for lift.
a) Well, if the hot-air balloon is accelerating downwards, it means the net force acting on it must be directed downwards. The only forces acting on the balloon are the gravitational force (which is downwards) and the lift force (which is upwards). So, the magnitude of the lift force must be equal to the magnitude of the gravitational force.
The formula for gravitational force is F = m * g, where F is the force, m is the mass, and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Therefore, the lift force acting on the hot-air balloon is equal to the mass of the balloon multiplied by the acceleration due to gravity.
Lift force = 720 kg * 9.8 m/s^2 ≈ 7056 N
b) Now, if the operator wants to accelerate upwards at 3.6 m/s^2, we need to find the total mass that needs to be thrown overboard in order to achieve this acceleration while keeping the lift force the same.
First, we need to calculate the new lift force required to counteract the increased weight due to the increased acceleration. The new lift force must be equal to the new gravitational force acting on the reduced mass.
The gravitational force is given by F = m * g, where F is the force, m is the mass, and g is the acceleration due to gravity (approximately 9.8 m/s^2).
So, the new gravitational force acting on the reduced mass is given by F = (720 kg - m) * g.
Since the lift force remains the same, we can equate the two expressions for force:
7056 N = (720 kg - m) * 9.8 m/s^2
Solving for m, we get:
m ≈ 720 kg - (7056 N / 9.8 m/s^2)
m ≈ 720 kg - 720 kg ≈ 0 kg
Well, it seems like our calculations show that we need to throw overboard all the mass in the hot-air balloon in order to achieve an upwards acceleration of 3.6 m/s^2 while keeping the lift force the same. So, in this case, the total mass that needs to be thrown overboard is 720 kg.
But hey, watch out for falling things! We don't want to create a clown shower!
Step 1: Find the lift force acting on the balloon.
Given:
Total mass of the balloon, m = 720 kg
Acceleration downwards, a = -2.7 m/s^2 (negative because it is in the opposite direction of the lift force)
To find the lift force, we can use Newton's Second Law: F = m * a.
Substituting the given values:
F = 720 kg *(-2.7m/s^2)
F = -1944 N (since the force is acting upwards, it has a negative sign)
Therefore, the lift force acting on the balloon is 1944 N.
Step 2: Find the total mass that needs to be thrown overboard.
Given:
Acceleration upwards, a = 3.6 m/s^2
Lift force, F = -1944 N (same as before)
To determine the total mass that needs to be thrown overboard, we need to find the change in mass required to achieve the desired acceleration.
Using Newton's Second Law again, we can rearrange the formula as follows: F = m * a.
Substituting the values:
-1944 N = m * 3.6 m/s^2
Solving for m:
m = -1944 N / 3.6 m/s^2
m ≈ -540 kg (since mass cannot be negative, we take the magnitude)
Therefore, approximately 540 kg of mass needs to be thrown overboard to achieve an upward acceleration of 3.6 m/s^2, assuming the lift force remains the same.
To find the lift force acting on the balloon, we can use Newton's second law of motion, which states that the force acting on an object is equal to its mass multiplied by its acceleration.
Given:
Total mass of the balloon (m) = 720 kg
Acceleration (a) = 2.7 m/s^2
a) Lift Force (F) = ?
Using Newton's second law: F = m * a
Substituting the given values:
F = 720 kg * 2.7 m/s^2
Calculating the lift force:
F = 1944 N
Therefore, the lift force acting on the balloon is 1944 Newtons.
b) To find the total mass that needs to be thrown overboard, we can rearrange the formula F = m * a to solve for mass.
Given:
Acceleration (a) = 3.6 m/s^2
Lift Force (F) = 1944 N
Rearranging the formula: m = F / a
Substituting the given values:
m = 1944 N / 3.6 m/s^2
Calculating the mass:
m ≈ 540 kg
Therefore, the total mass that needs to be thrown overboard is approximately 540 kg if the operator on the hot-air balloon decides to accelerate upwards at 3.6 m/s^2, assuming the upwards lift force remains the same.