Question

An object is thrown upward with a velocity of 32 ft/s from a stationary balloon which is 48 ft above the ground. If air resistance is ignored, the total time until the object impacts the ground is _____

Answer 1

Answer:

$t=2.98s$

Explanation:

First, we have to calculate the maximum height reached by the object. We use the equations of uniformly accelerated motion:

$y=y_0+v_0t-\frac{gt_1^2}{2}$

In order to calculate this, we have to know the time taken by the object to reach the maximum height:

$v_f=v_0-gt_1\\0=32\frac{ft}{s}-(32.15\frac{ft}{s^2})t_1\\t_1=\frac{-32\frac{ft}{s}}{-32.15\frac{ft}{s^2}}\\t_1=0.99s$

Now, we can calculate y:

$y=48ft+(32\frac{ft}{s})0.99s-\frac{(32.15\frac{ft}{s^2})(0.99s)^2}{2}\\y=63.93ft$

Now, we calculate the time taken by the object in free fall:

$y=y_0+v_0t+\frac{gt_2^2}{2}\\y=0ft+0\frac{ft}{s}t+\frac{gt_2^2}{2}\\t_2=\sqrt\frac{2y}{g}\\t_2=\sqrt\frac{2(63.93ft)}{32.15\frac{ft}{s^2}}\\t_2=1.99s$

Finally, adding $t_1$ and $t_2$, we get the total time until the object impacts the ground:

$t=t_1+t_2\\t=0.99s+1.99s\\t=2.98s$