During halftime of a basketball game a slingshot launches t shirts at the crowd. A T-shirt is launched from a height of 4 feet with an initial upward velocity of 88 feet per second. The T-shirt is caught 45 feet above the court. How long will it take the T-shirt to reach its maximum height? What is the maximum height? What is the range of the function that models the height of the T-shirt over time?
Normally in these type of questions, air resistance can be ignored, but launching a t-shirt ????
anyway ...
height = -16t^2 + 88t + 4
you want the vertex of the corresponding parabola
the x of the vertex is -88/(2(-16)) = 11/4
the corresponding height is -16(11/4)^2 + 88(11/4) + 4 = 125
but it was caught at a height of 45 ft (assuming on its way up)
so 45 = -16t^2 + 88t + 4
16t^2 - 88t + 41 = 0
solve using your favourite method of solving quadratics
and then interpret your result
Old book I guess. Assume a here = -32 ft/s^2 which is g in old units
h = 4 + 88 t - (32/2) t^2
v = 88 - 32 t
at max height v = 0
so t = 88/32 = 2.75 seconds
then at t = 2.75
h = 4 + 88 (2.75) - 16 (2.75)^2
= 4 + 242 - 121 = 125 feet at very top
now I suppose the T shirt is caught on the way down
at the top , h = 125, the speed is zero
so this is really something dropped from 125 ft and caught at 45 feet
fall 125 - 45 = 80 feet so you could calculate when it is caught.
Now the range is from a height of 4 feet to a height of 125 feet
To solve this problem, we can use the equations of motion for projectile motion. The equation of motion for the height of the T-shirt over time is h(t) = -16t^2 + v0t + h0, where h(t) is the height at time t, v0 is the initial velocity, and h0 is the initial height.
Let's calculate the time it will take for the T-shirt to reach its maximum height:
The initial upward velocity is 88 feet per second, so v0 = 88.
At its maximum height, the velocity of the T-shirt will be zero. Therefore, we can use the equation v(t) = v0 - 32t, where v(t) is the velocity at time t. Setting v(t) to zero and solving for t:
0 = 88 - 32t
32t = 88
t = 2.75 seconds
So, it will take the T-shirt 2.75 seconds to reach its maximum height.
Now, let's calculate the maximum height:
Plugging in the value of t into the equation for height:
h(t) = -16t^2 + v0t + h0
h(2.75) = -16(2.75)^2 + 88(2.75) + 4
h(2.75) = -16(7.5625) + 242 + 4
h(2.75) = -121 + 242 + 4
h(2.75) = 125 feet
Therefore, the maximum height of the T-shirt is 125 feet.
Finally, let's calculate the range of the function that models the height of the T-shirt over time:
The range is the horizontal distance the T-shirt travels before being caught. To find the range, we need to calculate the time it takes for the T-shirt to reach the height of 45 feet above the court.
Plugging in the value of h(t) = 45 into the equation for height and solving for t:
45 = -16t^2 + 88t + 4
-16t^2 + 88t - 41 = 0
Using the quadratic formula to solve for t:
t = (-b ± √(b^2 - 4ac)) / (2a)
t = (-88 ± √(88^2 - 4(-16)(-41))) / (2(-16))
t ≈ 2.06 seconds or t ≈ 4.69 seconds
Therefore, the T-shirt will take approximately 2.06 seconds or 4.69 seconds to reach the height of 45 feet.
The range is given by the difference between the two time values multiplied by the horizontal velocity, which is constant at 0 (since we are only concerned with vertical motion). Therefore, the range is 0 feet.
In this particular case, the T-shirt will not reach any horizontal distance before being caught, so the range of the function that models the height of the T-shirt over time is 0 feet.
To find the time it takes for the T-shirt to reach its maximum height, we can use the equation for vertical motion:
h = h0 + v0 * t - (1/2) * g * t^2
Where:
h = final height
h0 = initial height
v0 = initial velocity
g = acceleration due to gravity
t = time
In this case, the initial height (h0) is 4 feet, initial velocity (v0) is 88 feet per second, and acceleration due to gravity (g) is approximately 32.2 feet per second squared (assuming we neglect air resistance). The final height (h) at the maximum height is what we need to find.
At the maximum height, the T-shirt stops moving upward, so its final velocity will be 0. Therefore, we can set v = 0 and solve for t:
0 = v0 - g * t
Rearranging the equation, we have:
t = v0 / g
Plugging in the given values, we have:
t = 88 ft/s / 32.2 ft/s^2 ≈ 2.735 seconds
So, it will take approximately 2.735 seconds for the T-shirt to reach its maximum height.
To find the maximum height, we can substitute the time we just calculated (t = 2.735 seconds) into the equation for vertical motion:
h = h0 + v0 * t - (1/2) * g * t^2
Plugging in the values we have:
h = 4 ft + 88 ft/s * 2.735 s - (1/2) * 32.2 ft/s^2 * (2.735 s)^2
Simplifying the equation, we find:
h ≈ 4 ft + 240.08 ft - 119.735 ft
h ≈ 124.345 ft
So, the maximum height the T-shirt reaches is approximately 124.345 feet.
The range of the function that models the height of the T-shirt over time can be considered as the horizontal distance covered by the T-shirt during its flight. To find the range, we need to know how long the T-shirt was in the air.
First, let's calculate the time it takes for the T-shirt to reach the final height of 45 feet above the court. We can use the equation for vertical motion again and set h = 45 ft:
45 ft = 4 ft + 88 ft/s * t - (1/2) * 32.2 ft/s^2 * t^2
Rearranging and simplifying the equation, we get a quadratic equation:
16.1 t^2 - 88 t + 41 = 0
Solving this quadratic equation, we find two possible solutions:
t ≈ 0.915 seconds and t ≈ 5.1 seconds
Since the initial velocity is directed upwards, we can ignore the negative root and consider the positive root which gives us the time of flight.
Therefore, the time of flight is approximately 5.1 seconds.
Now we can find the range by multiplying the time of flight by the horizontal component of the velocity. Since the only given initial velocity is the vertical component, we need to find the horizontal component.
The initial vertical velocity (v0) of the T-shirt is 88 ft/s, and the angle of its trajectory is not given. Assuming a flat trajectory, the horizontal component of the velocity (v0x) is the same throughout the flight.
We can use the equation for horizontal motion:
range = v0x * t
Since the initial vertical velocity is given as 88 ft/s, we can use trigonometry to find the horizontal component:
v0x = v0 * cos(theta)
Now, we need to find theta, the angle between the initial velocity vector and the horizontal axis. Assuming a flat trajectory, theta is 90 degrees.
cos(90 degrees) = 0, so the horizontal component of the initial velocity becomes:
v0x = 88 ft/s * 0 = 0 ft/s
Therefore, the range of the function that models the height of the T-shirt over time is 0 feet.