Question Video: Finding the Distance between an Ascending Balloon and a Body Falling from It given the Balloon’s Velocity Mathematics

A balloon of mass 1086 kg was ascending vertically at 36 cm/s. If a body of mass 181 kg fell from it, find the distance between the balloon and the body 11 seconds after the body fell. Take 𝑔 = 9.8 m/sΒ².

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Video Transcript

A balloon of mass 1086 kilograms was ascending vertically at 36 centimeters per second. If a body of mass 181 kilograms fell from it, find the distance between the balloon and the body 11 seconds after the body fell. Take 𝑔 equal to 9.8 meters per second squared.

In order to answer this question, we will use the equations of motion or SUVAT equations together with Newton’s second law, which states that 𝐹 equals π‘šπ‘Ž, where 𝐹 is the sum of the forces, π‘š is the mass, and π‘Ž is the acceleration of the body. Let’s begin by considering the balloon. We are told it has a mass of 1086 kilograms. It was ascending vertically at 36 centimeters per second. As there are 100 centimeters in a meter, converting this to standard units, we have a velocity of 0.36 meters per second. At this stage, as the balloon is ascending at a constant velocity, we know that the acceleration is equal to zero meters per second squared.

There are two forces acting on the balloon in the vertical direction. Acting upwards, we have the left force, which we have labeled 𝐹, and acting downwards, we have the weight of the balloon. Taking the positive direction to be vertically upwards, we see that the sum of the forces is equal to 𝐹 minus 1086 multiplied by 9.8. This is equal to the mass 1086 multiplied by zero. The left-hand side is therefore simply equal to zero. Multiplying 1086 by 9.8 gives us 10642.8. And adding this to both sides of our equation, we have 𝐹 is equal to 10642.8. This is the left force generated by the hot air in newtons.

Next, we are told that a body of mass 181 kilograms falls from a balloon. Subtracting 181 from 1086 gives us 905. This means that the mass of the balloon is now 905 kilograms. Its weight is therefore equal to 8869 newtons. This is 905 multiplied by 9.8. Next, we can calculate the acceleration of the balloon by using the equation 𝐹 equals π‘šπ‘Ž once again. We have an upward force of 10642.8 newtons and a downward force of 8869 newtons. The difference between these values will be equal to 905π‘Ž. Dividing through by 905, we have π‘Ž is equal to 1773.8 divided by 905. This is equal to 1.96. The acceleration of the balloon after the body of mass falls from it is 1.96 meters per second squared.

We will now clear some space and consider the falling body. The body of mass 181 kilograms is accelerating vertically downwards. It will accelerate due to gravity. Therefore, π‘Ž is equal to 9.8 meters per second squared acting vertically downwards. Its initial speed is negative 0.36 meters per second as at the point it falls from a balloon, the balloon was moving with a constant velocity of 0.36 meters per second vertically upwards.

We are now in a position where we can calculate the distance between the balloon and the body 11 seconds after the body fell. We will do this using the equations of motion or SUVAT equations, where 𝑠 is the displacement in meters. 𝑒 and 𝑣 are the initial and final velocities in meters per second, π‘Ž is the acceleration in meters per second squared, and 𝑑 is the time in seconds. Considering the balloon where we take the positive direction to be vertically upwards, we have 𝑒 equals 0.36, π‘Ž equals 1.96, and 𝑑 equals 11. We will let the displacement or the distance traveled by the balloon be 𝑠 sub one.

The equation we will use is 𝑠 is equal to 𝑒𝑑 plus a half π‘Žπ‘‘ squared. Substituting in our values, we have 𝑠 sub one is equal to 0.36 multiplied by 11 plus a half multiplied by 1.96 multiplied by 11 squared. This is equal to 122.54. 11 seconds after the body falls from the balloon, the balloon has ascended 122.54 meters. We will now consider the body where the downward direction is considered positive. Our value of 𝑒 is negative 0.36, π‘Ž is equal to 9.8, and 𝑑 is equal to 11. We will let the displacement of the body be 𝑠 sub two. Substituting in our values to the same equation, we have 𝑠 sub two is equal to negative 0.36 multiplied by 11 plus a half multiplied by 9.8 multiplied by 11 squared. Typing the right-hand side into our calculator gives us 588.94. After 11 seconds, the body has fallen 588.94 meters.

The distance between the balloon and the body after 11 seconds is therefore equal to 122.54 plus 588.94. This is equal to 711.48 meters.

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