Question

A small bag of sand is released from an ascending hot-air balloon whose upward constant velocity is v0 = 1.95 m/s. Knowing that at the time of the release the balloon was 57.8 m above the ground, determine the time, τ, it takes the bag to reach the ground from the moment of its release.

Answer 1

@Shalante

Answer 2

When the bag leaves the balloon, the only force acting on it is gravity. We know the following information: initial velocity (vi), distance (d), acceleration (a). We want to calculate time (t). The equation that uses all these variables is:

\[d=v_it+\frac{1}{2}at^2\]

Answer 3

\[57.8m =(1.95 m/s)(t) +\frac{ 1 }{ 2 }(-9.81 m/s ^{2})(t ^{2})\]

Answer 4

But how do I solve it exactly? As in getting the time all to one side and such?

Answer 5

@Data_LG2
@whpalmer4
@matt101

Answer 6

do you know how to solve a quadratic formula? you have to use the same method for this problem

Answer 7

It's been awhile since I've done that.

Answer 8

\(\sf 57.8m =(1.95 m/s)(t) +\frac{ 1 }{ 2 }(-9.81 m/s ^{2})(t ^{2})\\
-4.91t ^{2} + 1.95t- 57.8 =0 \)

Use the quadratic formula to solve for t:

\(\sf \Large t=\frac{ -b \pm \sqrt{b^2-4ac} }{2a}\)

where a= -4.9, b=1.95, c= -57.8
you just have to plug these value in the formula to calculate for t.
you must get two values: one negative and one positive value.
But in this case, we must ignore the negative value since there is no negative time...

Answer 9

I got 4.78s

Answer 10

Is that the wrong answer?

Answer 11

I think I made a mistake. Displacement is towards down so it should be -57.8m
hence, the equation must be:

\(\sf -57.8m =(1.95 m/s)(t) +\frac{ 1 }{ 2 }(-9.81 m/s ^{2})(t ^{2})\\
-4.91t ^{2} + 1.95t +57.8 =0\)

you will get an imaginary number if you didn't follow the right sign convention. You already used -9.8 m/s^2, which means your that your upward direction is your positive.

Answer 12

So am I needing to switch the -57.8 to a positive and plug it back in into the quadratic equation?

Answer 13

yes

Answer 14

I get -77.82 that way.

Answer 15

no.. can you show me your solution?

Answer 16

\[-1.95+\sqrt{(1.95^{2}-4(-4.9)(57.8)}/2(-4.9)\]

Answer 17

You divide the entire thing by 2 (-4.9)

Answer 18

remember that a quadratic can have two solutions, as I mentioned above, you are expected to get a negative and a positive value. Notice that in the quadratic formula that it has "\(\sf \color{red} \pm\)" symbol which means to add and subtract. You just did the "+" part.. you also have to do this \(\sf \large \frac{-1.95 \color{red}{-}\sqrt{(1.95^{2})-4(-4.9)(57.8)}\ }{2(-4.9)}\)

Answer 19

87.38s

Answer 20

I got 3.64s. Maybe you made some mistakes on your calculation, try checking it again and let me know if you already got it :)

Answer 21

I keep getting the 87.38

Answer 22

okay, let's do it step by step
Inside the radical first: 1.95^2 = 3.8025
then -4(-4.9)(57.8)= 1132.88

add those two together 1136.6825
if you get the square root of it, what will you get?

Answer 23

I got it finally!

Answer 24

3.6392

Answer 25

yay \.^_^./ just don't forget the unit.

Answer 26

Right. Thanks for helping on this. Do you have time to help with another?

Answer 27

not sure though, i'm doing my physics lab report. Just post it, maybe other users who are good with physics can help you too :)

Answer 28

Alright. Thanks!