Final answer:
To calculate the time it takes for the sandbag to reach the ground, we must first determine the time to reach the highest point using the initial velocity and the rate of gravitational acceleration. We then calculate the time for the bag to fall from the highest point to the ground. The total time is the sum of both durations.
Step-by-step explanation:
To determine the time τ it takes for a bag of sand to reach the ground after being released from an ascending hot-air balloon, we can use the kinematic equations of motion. The initial velocity of the bag of sand is given as v₀ = 2.95 m/s upward. Since the bag of sand is subject to gravity, its upward velocity will decrease at a rate of g = 9.81 m/s² until it stops and then starts falling down towards the ground.
We can calculate the time it takes to reach the highest point using the formula v = v₀ - gt, where v is the final velocity (0 m/s at the highest point). Solving for t gives us the time to reach the highest point: tup = v₀ / g. After reaching the highest point, the sandbag will fall under gravity alone. We can then apply the formula for the distance covered during free fall d = 1/2gt² to find out how long it takes to fall the remaining distance. We need to consider the total distance (the initial height plus the additional height gained while the bag was still moving upward).
The total time taken (τ) to reach the ground will be the sum of the time taken to stop and the time taken to fall. The problem requires the use of quadratic equations to solve for tdown and, consequently, τ.