An object that is projected straight downward
with initial velocity v feet per second travels a
distance s vt 16t
2
=+ , where t =time in seconds.
If Ramón is standing on a balcony 84 feet above
the ground and throws a penny straight down
with an initial velocity of 10 feet per second, in
how many seconds will it reach the ground?
2. h = Vo*t + 0.5g*t^2 = 84 Ft.
10t + 16t^2 = 84
16t^2 + 10t - 84 = 0
Use Quadratic Formula.
t = 2 s.
Henry is right
To find the time it takes for the penny to reach the ground, we need to find the value of t when the distance traveled is equal to the distance from the balcony to the ground.
Given:
Initial velocity of the penny, v = 10 ft/s
Distance from balcony to ground, s = 84 ft
Using the provided equation: s = vt + 16t^2
Substituting the given values: 84 = 10t + 16t^2
Rewriting the equation: 16t^2 + 10t - 84 = 0
This is a quadratic equation in standard form. To solve for t, we can use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
Where:
a = 16
b = 10
c = -84
Substituting these values into the quadratic formula:
t = (-10 ± √(10^2 - 4(16)(-84))) / (2(16))
Simplifying further:
t = (-10 ± √(100 + 5376)) / 32
t = (-10 ± √5476) / 32
t = (-10 ± 74) / 32
Now we have two possible values for t:
t1 = (-10 + 74) / 32 = 64 / 32 = 2
t2 = (-10 - 74) / 32 = -84 / 32 = -2.625
Since time cannot be negative in this case, we can discard the negative value.
Therefore, the penny will reach the ground in approximately 2 seconds.
To answer this question, we need to find the time at which the object reaches the ground.
Given that Ramón throws the penny straight down with an initial velocity of 10 feet per second, the equation that represents the distance traveled by the object is s = 84 + 10t + 16t^2.
The distance traveled is the height of the balcony (84 feet) plus the distance traveled by the penny during the time it takes to reach the ground.
Since the penny reaches the ground, the distance traveled is equal to 0 at that time. Therefore, we can set the equation equal to 0:
84 + 10t + 16t^2 = 0
Now, we need to solve this quadratic equation to find the value of t. We can use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
In our equation, a = 16, b = 10, and c = 84. Substituting these values into the formula:
t = (-10 ± √(10^2 - 4 * 16 * 84)) / (2 * 16)
Simplifying further:
t = (-10 ± √(100 - 5376)) / 32
t = (-10 ± √(-5276)) / 32
Since √(-5276) is imaginary, the penny will never reach the ground.