For Questions 12-12, during halftime of a football game, a slingshot launches T-shirts at the crowd. A T-shirt is launched from a height of 3 feet with an initial upward velocity of 72 feet per second. The T-shirt is caught 50 feet above the field. What is the maximum height of the T-shirt? What is the range of the function?
What angle was the shot? I assume it was not shot straight up.
So, unless you provide the angle of launch, or the distance to the catcher, there is no way to pin down the max height or the range.
Given a launch angle of θ, you know the height at a horizontal distance x is
h = 3 + tanθ x - 16/(72 cosθ)^2 x^2
So, either you know x when h=50
or you need to know the angle θ
Then recall that the range is
R = 72^2/16 sin2θ
the relevant free fall equation is ... h = -16 t^2 + 72 t + 3
max height is on the axis of symmetry of the parabola (quadratic)
... tmax = -72 / (2 * -16)
... plug tmax back into the equation to find hmax
the range is from the ground to hmax
Nice catch, Scott. I neglected to read the word "upward."
And I also mistook the range of the function to be the range of the missile.
well from 3 feet to max h
Y = Yo + g*Tr = 0.
72 + (-32)Tr = 0,
Tr = 2.25 s. = Rise time.
a. Y^2 = Yo^2 + 2g*h = 0.
72^2 + (-64)h = 0,
h = 81 Ft. above launching point.
hmax = 81 + 3 = 84 Ft. above gnd.
h = 0.5g*Tf^2 = 50.
16Tf^2 = 50,
Tf = 1.77 s. = Fall time.
Range = Xo * (Tr+Tf) = Xo * (2.25+1.77).
Yo = Vo*sinA = 72.
Xo = Vo*CosA.
To find the range , I'll need to know the initial velocity(Vo) or the launch angle(A).
To find the maximum height of the T-shirt, we need to determine the peak point in its trajectory.
First, let's find the time it takes for the T-shirt to reach its highest point. We can use the equation:
h = h0 + v0t - 16t^2
Where:
h is the height at any given time (t)
h0 is the initial height (3 feet)
v0 is the initial velocity (72 feet per second)
t is the time in seconds
Since we want to find the time when the height is at its maximum, we can assume that the velocity at this point is 0. So we can rewrite the equation as:
0 = 3 + 72t - 16t^2
Now, we can solve this quadratic equation for t. Setting it equal to zero gives us:
16t^2 - 72t - 3 = 0
Using the quadratic formula (t = (-b ± √(b^2 - 4ac)) / (2a)), we find that the solutions are t ≈ 4.03 and t ≈ 0.186.
Since the T-shirt is initially launched upward, we can exclude the negative solution. Therefore, the time it takes for the T-shirt to reach its highest point is approximately 0.186 seconds.
Now, let's find the maximum height by substituting this time back into the original equation:
h = h0 + v0t - 16t^2
h = 3 + 72(0.186) - 16(0.186)^2
h ≈ 15.62 feet
So, the maximum height of the T-shirt is approximately 15.62 feet.
To find the range of the function, we need to determine the horizontal distance the T-shirt travels. The range can be obtained by multiplying the horizontal velocity (which remains constant) by the total time of flight.
The horizontal distance can be found using the equation:
range = horizontal velocity * time
The horizontal velocity remains constant at 72 feet per second, and the total time of flight can be calculated as twice the time it takes for the T-shirt to reach its maximum height:
total time of flight = 2 * 0.186 = 0.372 seconds
Substituting these values into the equation, we get:
range = 72 * 0.372
range ≈ 26.784 feet
Therefore, the range of the function is approximately 26.784 feet.