Question

A hot-air balloon rises vertically at a constant rate of 8 m/s. A second hot-air balloon descends passed the first hot-air balloon at a rate of 12 m/s relative to the first hot-air balloon. What is the velocity of the second hot-air balloon relative to the ground?

Answer 1

One

When moving in opposite directions the two velocities are added together to give the final velocity.

But you have the complicating factor of what speed the downward balloon is actually going. Once you solve that, you can get the actual answer.

They appear to be moving away from each other at a total speed of 12 m/s when either of the balloons is the point of view. The person on the ground sees what actually is happening. He sees one going up at 8 m/s and the other going down at 4 m/s.

That last number is your answer

Two

The Dodge is moving forward. The BMW appears to be moving backward. It isn't. It is moving forward as well at a rate of 63 mph.

Three

The ground viewer sees both of them going up. The faster balloon (8 m/s) is actually moving away from the slower one at a rate of 3 m/s according to the person looking at the situation on the ground. 8 -3 is what the ground viewer sees.

Four

The package looks like it is going an additional 3.1 m/s added onto the 9.5 m/s

So the answer is 12.6 down

Answer 2

In every case, you add the speed of the object (relative to the reference point) to the speed of the reference point (relative to the ground) to find the speed of the object relative to the ground. (We choose to use + to mean "up" or "forward".)

1. The speed of the 2nd balloon relative to the first is -12 m/s. The speed of the first relative to the ground is +8 m/s. Thus the speed of the 2nd balloon relative to the ground is

... (-12 m/s) + (+8 m/s) = -4 m/s . . . . 4 m/s, down

2. The speed of the Dodge relative to the ground is 70 mph. The speed of the BMW relatvie to the Dodge is -7 mph, so the speed of the BMW relative to the ground is

... (-7 mph) + (+70 mph) = 63 mph

3. The speed of the 2nd balloon relative to the first is -3 m/s. The speed of the first is +8 m/s relative to the ground, so the speed of the 2nd balloon relative to the ground is

... (-3 m/s) + (+8 m/s) = +5 m/s . . . . 5 m/s, up

4. The speed of the package relative to the helicopter is -9.5 m/s. The speed of the helicopter relative to the ground is -3.1 m/s, so the speed of the package relative to the ground is

... (-9.5 m/s) + (-3.1 m/s) = -12.6 m/s . . . . 12.6 m/s, down