Given a Constant Acceleration magnitude of g/4, Find the value of t

[baumbad]

Homework Statement

A hot-air balloon takes off from the ground traveling vertically with a constant upward acceleration of magnitude g/4. After a time interval Δt, a crew member releases a ballast sandbag from the basket attached to the balloon.
How many seconds does it take the sandbag to reach the ground? Express your answer in terms of Δt.

Relevant Equations

s = ut + 1/2at^(2)

I got to the quadratic equation of the motion where: 4gt^(2) - g(delta t)t - g(delta t) = 0 and tried to solve for t. In this case, we would take the positive discriminate since we are dealing with the passing of time.

t = ((sqrt(17) g(delta t)) + g (delta t)) / (8g)

However, this is the wrong answer and I am not sure why.

 

[haruspex]

4gt^(2) - g(delta t)t - g(delta t) = 0

Please show the steps by which you got that.
What was the velocity of the sandbag when it was released?

 

[baumbad]

this is the work that I used, however, I'm not sure that this is the correct approach. At a particular time, the sandbag goes from an upward acceleration of g/4 to free fall.

 

[kuruman]

Your approach looks correct. You have two equations with arrows pointing at them. Check how you get from the first one to the second.

 

[baumbad]

Thank you very much for your help!

 

[haruspex]

Also check the signs in the final step.

 

[jbriggs444]

After a time interval Δt, a crew member releases a ballast sandbag from the basket attached to the balloon.
How many seconds does it take the sandbag to reach the ground? Express your answer in terms of Δt.

Note that the question should have asked "how much time", not "how many seconds".

As written, it is a bit tricky to pick the right units for the required answer.