Let's go through each part of the question step by step:
a. To find when the projectile's height above the ground is 1024 ft, we can use the equation for vertical displacement of a projectile:
y = v0*t - (1/2)*g*t^2
where:
y is the height above the ground,
v0 is the initial velocity,
g is the acceleration due to gravity (32 ft/s^2),
and t is the time.
Plugging in the given values, we have:
1024 = 320*t - (1/2)*32*t^2
Simplifying, we get a quadratic equation:
16t^2 - 320t + 1024 = 0
Factoring the equation, we have:
16(t - 8)(t - 8) = 0
Solving for t, we find that t = 8 seconds.
Therefore, the projectile's height above the ground will be 1024 ft at 8 seconds.
b. To find when the height of the projectile is at least 24 ft, we need to determine the time it takes for the projectile to reach its highest point. At this point, the velocity will be 0. Using the equation for vertical velocity, we have:
v = v0 - g*t
Setting v = 0, we can solve for t:
0 = 320 - 32*t
Solving for t, we find that t = 10 seconds.
Since the projectile starts from the ground, it will pass the 24 ft mark on its way up, so the height of 24 ft will be reached before 10 seconds.
c. The time for the object to strike the ground can be found by setting the height, y, equal to 0. Using the equation for vertical displacement, we have:
0 = 320t - (1/2)*32*t^2
This simplifies to:
16t^2 - 320t = 0
Factoring out t, we find:
16t(t - 20) = 0
So t = 0 or t = 20 seconds.
Since the object was launched from the ground, we can discard t = 0. Thus, the object will strike the ground at 20 seconds.
d. The maximum height of the projectile can be found by determining the time it takes for the vertical velocity to reach zero. At this point, the projectile is at its highest point. Using the equation for vertical velocity, we have:
v = v0 - g*t
Setting v = 0, we can solve for t:
0 = 320 - 32*t
Solving for t, we find that t = 10 seconds.
Plugging this value of t into the equation for vertical displacement, we have:
y = 320*10 - (1/2)*32*(10)^2
Simplifying, we get:
y = 1600 ft.
Therefore, the maximum height of the projectile is 1600 ft.
To summarize:
a. The projectile's height above the ground will be 1024 ft at 8 seconds.
b. The height of the projectile will be at least 24 ft before it reaches its highest point at 10 seconds.
c. The object will strike the ground at 20 seconds.
d. The maximum height of the projectile is 1600 ft.