A small bag of sand is released from an ascending hot-air balloon whose upward constant velocity is v0 = 1.45 m/s. Knowing that at the time of the release the balloon was 70.3 m above the ground, determine the time, τ, it takes the bag to reach the ground from the moment of its release.

Well you need to use the equations of motion

v=u+at
v^2=u^2+2aS and
S=ut+(1/2)at^2

a = -g in this question

What happens first is the bag moves upwards for a short time (as it has a velocity of 1.45m/s) and then falls down

You can do this question in one go (instead of working out how long it takes to go up and then how long it takes to fall down and add the two)

Use the following
u (initial velocity) = 1.45m/s
S (displacement) = -70.3m
a (acceleration) = -g = -9.8m/s

Use those to find t using the equations above (you need only 1 of them)

To solve this problem, we can use the equations of motion for the bag of sand.

First, we need to find the time it takes for the bag of sand to reach the ground. We can use the equation:

h = v0 * t + (1/2) * g * t^2

where h is the height of the bag above the ground, v0 is the initial velocity of the balloon (1.45 m/s), g is the acceleration due to gravity (9.8 m/s^2), and t is the time it takes for the bag to reach the ground.

Given that the initial height of the balloon above the ground is 70.3 m (h = 70.3 m), we can rearrange the equation to solve for t:

(1/2) * g * t^2 + v0 * t - h = 0

Substituting the known values, we have:

(1/2) * 9.8 * t^2 + 1.45 * t - 70.3 = 0

This is a quadratic equation in terms of t. We can solve it using the quadratic formula:

t = (-b ± √(b^2 - 4ac)) / (2a)

where a = (1/2) * g, b = v0, and c = -h.

Plugging in the values, we have:

t = (-(1.45) ± √((1.45)^2 - 4 * (1/2) * 9.8 * (-70.3))) / (2 * (1/2) * 9.8)

Simplifying further:

t = (-1.45 ± √(2.1025 + 1376.72)) / 9.8

t = (-1.45 ± √1378.8225) / 9.8

Now we can calculate the values inside the square root first:

√1378.8225 ≈ 37.139

Substituting back into the equation and calculating separately for "+" and "-", we have:

t1 = (-1.45 + 37.139) / 9.8 ≈ 3.582 s (positive value)

t2 = (-1.45 - 37.139) / 9.8 ≈ -4.067 s (negative value)

Since time cannot be negative, we take the positive value as the time it takes for the bag to reach the ground.

Therefore, the time it takes for the bag of sand to reach the ground is approximately 3.582 seconds.