A research balloon of total mass 225 kg is descending vertically with a downward acceleration of 1.4 m/s2. How much ballast must be thrown from the car to give the balloon an upward acceleration equal to 3.6 m/s2, presuming that the upward lift of the balloon does not change.
Well, if the balloon wants to uplift its game and accelerate upwards, it needs to drop some weight. It's like going on a diet but with a bit more physics involved.
To find out how much ballast needs to be thrown from the car, we first need to calculate the total force required. We know that force equals mass multiplied by acceleration.
The downward force acting on the balloon is given by the equation F = m * a, where F is the force, m is the mass, and a is the acceleration. So, F = (225 kg) * (1.4 m/s^2).
Now, in order to give the balloon an upward acceleration of 3.6 m/s^2, we need to find the total force necessary. This force is given by F = m * a, where F is the force, m is the mass, and a is the acceleration. So, F = (225 kg) * (3.6 m/s^2).
To find the amount of ballast that needs to be thrown, we subtract the initial force from the final force. So, we have (225 kg) * (3.6 m/s^2) - (225 kg) * (1.4 m/s^2).
Now, let's do some number-crunching and find out how much ballast should be ditched. I hope the answer makes you lighter, just like a good joke!
(Pulls out calculator and starts calculating...after some calculations)
The amount of ballast that needs to be thrown from the car is approximately 472.5 kg. That's right, 472.5 kg! So, start cleaning out the car and get ready to say goodbye to some serious weight. Maybe you'll even find that missing SOCK(et) in the process!
To find the amount of ballast that must be thrown from the car, we need to consider two forces acting on the balloon: the weight of the balloon and the upward lift.
1. Calculate the weight of the balloon:
The weight of an object can be calculated by multiplying its mass by the acceleration due to gravity. In this case, the weight (W) of the balloon is given by W = mass x acceleration due to gravity. The acceleration due to gravity is approximately 9.8 m/s^2.
Given:
Mass of the balloon (m) = 225 kg
Acceleration due to gravity (g) = 9.8 m/s^2
W = m x g
W = 225 kg x 9.8 m/s^2
W ≈ 2205 N
2. Determine the necessary ballast:
To calculate the required amount of ballast, we need to consider the net force acting on the balloon. The net force is the difference between the upward lift force and the weight of the balloon.
Net force (F_net) = Lift force (F_lift) - Weight (W)
Given:
Initial downward acceleration (a_initial) = -1.4 m/s^2
Final upward acceleration (a_final) = 3.6 m/s^2
For the initial descent, the net force is given by:
F_net_initial = F_lift - W = m x a_initial
For the final ascent, the net force is given by:
F_net_final = F_lift - W = m x a_final
Since the upward lift force remains constant, F_lift is the same in both cases.
So, equating the two net forces, we have:
m x a_initial = m x a_final
Solving for the required mass to be thrown (m_throw):
m_throw = m x (a_final - a_initial)
Given:
a_initial = -1.4 m/s^2
a_final = 3.6 m/s^2
m_throw = 225 kg x (3.6 m/s^2 - (-1.4 m/s^2))
m_throw = 225 kg x 5 m/s^2
m_throw = 1125 kg⋅m/s^2
Therefore, approximately 1125 kg of ballast must be thrown from the car to give the balloon an upward acceleration of 3.6 m/s^2, assuming the upward lift of the balloon does not change.