Solve Balloon Problem Homework: Time & Speed Impact

[CrossFit415]

Homework Statement

Question: A hot-air balloon is ascending at the rate of 12m/s and is 80m above the ground when a package is dropped over the side.
a.)How long does the package take to reach the ground? b.)With what speed does it hit the ground?

Homework Equations

I know -- acceleration = -9.8 Initial velocity = 12 m/s and initial height = 80 m.

I know I have to use this formula
Xf = Vo + 1/2 (g) (T)²

Vo = 12 m/s
Xf = 80 m
g = -9.8 m/s²

I set Xf to 0.How would I find Time?

Thanks

 

[Doc Al]

I know I have to use this formula
Xf = Vo + 1/2 (g) (T)²

I think you mean:
Xf = Xo + Vo(T) - 1/2 (g) (T)²
where g = 9.8 m/s²

How would I find Time?

Time is the only unknown in the above quadratic equation, so solve for T.

 

[CrossFit415]

How would I go on about doing that? I can't seem to get T by itself in this equation.

 

[NasuSama]

I think you mean:
Xf = Xo + Vo(T) - 1/2 (g) (T)²
where g = 9.8 m/s²

Time is the only unknown in the above quadratic equation, so solve for T.

That user is correct by the way because this is the form you should get. Well, since the object starts 80 m above the ground, x_0 = 80 obviously.

Here is the equation you get:

x_f = 80 + 12t - 4.9t²

How would I go on about doing that? I can't seem to get T by itself in this equation.

It's not impossible to find t. To find t, use the quadratic equation as that user indicates. That is the way to find t. Remember that:

at² + bt + c = 0 OR c + bt + at² = 0

t = (-b ± √(b² - 4ac))/(2a)

OR

t = (-b + √(b² - 4ac))/(2a) or t = (-b - √(b² - 4ac))/(2a)

By letting the corresponding values be the a, b, c variables and then, solving for t, you should get the answer (it must be positive!).

 

[Nickie]

To find the time it takes for the package to reach the ground, you can use the formula Xf = Xo + Vot + 1/2at^2, where Xf is the final position (ground level in this case), Xo is the initial position (80m above the ground), Vo is the initial velocity (12m/s), a is the acceleration due to gravity (-9.8m/s^2), and t is the time. Plugging in the values, we get 0 = 80 + 12t + 1/2(-9.8)t^2. Rearranging and solving for t, we get t = 3.66 seconds.

To find the speed at which the package hits the ground, you can use the formula Vf = Vo + at, where Vf is the final velocity (which is the speed at which the package hits the ground), Vo is the initial velocity (12m/s), a is the acceleration due to gravity (-9.8m/s^2), and t is the time (which we found to be 3.66 seconds). Plugging in the values, we get Vf = 12 + (-9.8)(3.66) = -35.85m/s. This means that the package will hit the ground with a speed of 35.85m/s.

Please note that the negative sign indicates that the package is moving downwards, which is in the opposite direction of the initial velocity.