An object is released from the top of a building 320 ft high. The initial velocity is 16 ft/s. How many seconds later will the object hit the ground?
did anyone get the answer :/
To find out how many seconds later the object will hit the ground, we can use the equation of motion for free fall:
h = (1/2)gt^2 + v0t + h0
Where:
h is the height (in this case, it is the initial height of 320 ft)
g is the acceleration due to gravity (approximately 32.2 ft/s^2)
t is the time (what we want to find)
v0 is the initial velocity (16 ft/s)
h0 is the initial height (320 ft)
Since we want to find the time when the object hits the ground, we can set h to zero:
0 = (1/2)(32.2)t^2 + 16t + 320
Now, we can solve this quadratic equation for t.
To find the time it takes for the object to hit the ground, we can use the equation of motion:
\(h = v_0t + \frac{1}{2}gt^2\)
Where:
- \(h\) is the height (320 ft)
- \(v_0\) is the initial velocity (16 ft/s)
- \(g\) is the acceleration due to gravity (approximately 32 ft/s²)
- \(t\) is the time
Rearranging the equation, we get:
\(\frac{1}{2}gt^2 + v_0t - h = 0\)
This is a quadratic equation in terms of \(t\). We can solve for \(t\) using the quadratic formula:
\(t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
In this equation, \(a = \frac{1}{2}g\), \(b = v_0\), and \(c = -h\).
Substituting these values and solving the equation, we can find the time it takes for the object to hit the ground.
Let's calculate it step by step:
\(a = \frac{1}{2}g = \frac{1}{2} \times 32 = 16\) ft/s²
\(b = v_0 = 16\) ft/s
\(c = -h = -320\) ft
Substituting these values into the quadratic formula:
\(t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
\(t = \frac{-16 \pm \sqrt{(16)^2 - 4 \times 16 \times (-320)}}{2 \times 16}\)
\(t = \frac{-16 \pm \sqrt{256 + 20480}}{32}\)
\(t = \frac{-16 \pm \sqrt{20736}}{32}\)
\(t = \frac{-16 \pm 144}{32}\)
Now, solve for both possible values of \(t\):
\(t_1 = \frac{-16 + 144}{32} = \frac{128}{32} = 4\) seconds
\(t_2 = \frac{-16 - 144}{32} = \frac{-160}{32} = -5\) seconds
Since time cannot be negative, the object will hit the ground after 4 seconds.