a hot air balloon rise above the ground, accelerating upwards at a rate of 2 m/s^2. The total mass of the hot air balloon and it occupants is 400kg. the density of the air is 1.3 kg/m^3.
a) Determine the volume if the hot air balloon.
b) if this acceleration is maintained for 5 seconds, how high will the balloon rise?
a) The probably want you to use
Force = Mass* aaceleration
to get the net force on the balloon, and then use
Net Force = Buoyancy - Weight
to get the buoyancy force. Once you have that, you can get the volume of the balloon using Archimedes Principle.
This approach neglects the atmospheric drag force, which might be appreciable, but perhaps not for the first 5 seconds.
b) Distance risen = (1/2) a t^2
(if a remains constant)
ur wrong
To calculate the volume of the hot air balloon, we need to use Archimedes' principle, which states that the buoyant force experienced by an object submerged in a fluid is equal to the weight of the fluid displaced by the object.
a) Determine the volume of the hot air balloon:
The buoyant force (F_b) can be calculated by multiplying the density of the fluid (ρ) by the volume of the fluid displaced (V) and the acceleration due to gravity (g), as given by the formula: F_b = ρ * V * g.
In this case, the buoyant force is equal to the weight of the hot air balloon and its occupants, which is given by the formula: F_g = m * g, where m is the total mass and g is the acceleration due to gravity.
Since the balloon is accelerating upwards at 2 m/s^2, the net force acting on it (F_net) can be found by subtracting the buoyant force from the weight: F_net = F_g - F_b = m * g - ρ * V * g.
Given:
Total mass (m) = 400 kg
Density of air (ρ) = 1.3 kg/m^3
Acceleration due to gravity (g) = 9.8 m/s^2 (approximate value)
Using the given values, we can substitute the values into the formula for F_net and solve for the volume (V).
F_net = m * g - ρ * V * g
F_net = (400 kg) * (9.8 m/s^2) - (1.3 kg/m^3) * V * (9.8 m/s^2)
F_net = 3920 N - 12.74 V N
To find the volume (V), we set F_net equal to zero (since the balloon has reached equilibrium), resulting in:
0 = 3920 N - 12.74 V N
12.74 V N = 3920 N
V = (3920 N) / (12.74 N)
V ≈ 308 m^3
Therefore, the volume of the hot air balloon is approximately 308 m^3.
b) To determine how high the balloon will rise if the acceleration is maintained for 5 seconds, we can use the kinematic equation:
s = ut + (1/2)at^2
Given:
Initial velocity (u) = 0 m/s (since the balloon starts from rest)
Acceleration (a) = 2 m/s^2
Time (t) = 5 seconds
Plugging in the values, we get:
s = (0 m/s) * (5 s) + (1/2) * (2 m/s^2) * (5 s)^2
s = 0 m + (1/2) * 2 m/s^2 * 25 s^2
s = 0 m + 1 m/s^2 * 25 s^2
s = 25 m/s^2
Therefore, if the acceleration is maintained for 5 seconds, the balloon will rise to a height of 25 meters.