An arrow is shot vertically upward from the top of a cliff 80 feet tall, with an initial velocity of 64 ft/s, (the acceleration due to gravity is -16ft/sec).
a. Write the quadratic equation representing this scenario when h is 0.
b. Find the roots (solutions) for this quadratic equation, solving by factoring.
assuming the quadratic equation represents the height of the arrow above the ground at the base of the cliff (and that the shot is made just past the edge of the cliff, so the falling arrow does not hit the cliff), then
the height h as a function of time is
h(t) = 80 + 64t - 16t^2
h=0 when the arrow hits the ground at the base of the cliff.
the rest should be no sweat
Thanks Steve
To solve this problem, let's first consider the motion of the arrow. When the arrow is shot vertically upward, it experiences acceleration due to gravity. The gravitational acceleration is given as -16 ft/s^2 because it acts in the opposite direction to the arrow's motion.
a. Write the quadratic equation representing this scenario when h is 0:
To represent the motion of the arrow, we can use the equation of motion:
h = h0 + v0t + (1/2)at^2,
where:
h represents the height of the arrow at time t
h0 represents the initial height of the arrow (80 ft)
v0 represents the initial velocity of the arrow (64 ft/s)
a represents the acceleration due to gravity (-16 ft/s^2)
t represents the time
When the arrow reaches the ground, its height (h) will be 0. Therefore, we can write the equation as:
0 = 80 + 64t - 16t^2.
b. Find the roots (solutions) for this quadratic equation, solving by factoring:
To find the roots of the quadratic equation, we need to set the equation equal to zero:
-16t^2 + 64t + 80 = 0.
Now, let's factor the equation:
-16t^2 + 64t + 80 = 0
-16(t^2 - 4t - 5) = 0.
The factored form can be simplified further:
-16(t - 5)(t + 1) = 0.
Now, set each factor equal to zero and solve for t:
t - 5 = 0 (or) t + 1 = 0.
Solving these equations gives us the roots:
t = 5 (or) t = -1.
Therefore, the roots (solutions) for the quadratic equation are t = 5 and t = -1.