height of a projectile thrown straight up from top of 480 ft hill t seconds after bing thrown up with initial velocity vo ft/s is
h = -16t^2 +vot + 480
If vo = 96, find times when the projectile will be at a) height of 592 ft above the ground and b) crash on the ground.
so h = -16t^2 + 96t + 480
a)
you want
-16t^2 + 96t + 480 = 592
t^2 - 6t + 7 = 0
(t-7)(t+1) =0
t = 7 or a negative
b)
same thing except now you want
-16t^2 + 96t + 480 = 0
solve for t, use only the positive result
To find the time when the projectile will be at a height of 592 ft above the ground, we can substitute the given values into the equation h = -16t^2 + vot + 480 and solve for t.
a) Height of 592 ft above the ground:
Substituting h = 592 and vo = 96 into the equation, we get:
592 = -16t^2 + 96t + 480
Rearranging the equation:
16t^2 - 96t + 480 - 592 = 0
16t^2 - 96t - 112 = 0
Dividing the equation by 16 to simplify:
t^2 - 6t - 7 = 0
(t - 7)(t + 1) = 0
Setting each factor equal to zero:
t - 7 = 0 or t + 1 = 0
t = 7 or t = -1
Since time cannot be negative in this context, the projectile will be at a height of 592 ft above the ground at t = 7 seconds.
b) To find the time when the projectile crashes on the ground, we need to find the time when h = 0.
Substituting h = 0 and vo = 96 into the original equation, we get:
0 = -16t^2 + 96t + 480
Rearranging the equation:
16t^2 - 96t - 480 = 0
Dividing the equation by 16 to simplify:
t^2 - 6t - 30 = 0
Using the quadratic formula, we can solve for t:
t = (-(-6) ± √((-6)^2 - 4(1)(-30))) / 2(1)
t = (6 ± √(36 + 120)) / 2
t = (6 ± √156) / 2
t ≈ (6 ± 12.49) / 2
Since time cannot be negative in this context, the only valid solution is:
t ≈ (6 + 12.49) / 2
t ≈ 18.49 / 2
t ≈ 9.25
Therefore, the projectile will crash on the ground at approximately t = 9.25 seconds.
To find the times when the projectile will be at a height of 592 ft and when it will crash on the ground, we need to solve the equation for the value of t.
a) Height of 592 ft above the ground:
Given that h = 592, we can substitute this value into the equation and solve for t:
592 = -16t^2 + 96t + 480
This equation is a quadratic equation, so to solve it, we can make it equal to zero by subtracting 592 from both sides:
0 = -16t^2 + 96t + 480 - 592
Simplifying,
0 = -16t^2 + 96t - 112
Now, we can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. Let's use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
In our case, a = -16, b = 96, and c = -112. Substituting these values into the quadratic formula:
t = ( -96 ± √(96^2 - 4(-16)(-112)) ) / (2(-16))
Simplifying further,
t = ( -96 ± √(9216 - 7168) ) / (-32)
t = ( -96 ± √(2048) ) / (-32)
Now, let's calculate the square root of 2048:
√2048 ≈ 45.25
So our equation becomes:
t = ( -96 ± 45.25 ) / -32
For t = ( -96 + 45.25 ) / -32, we get t ≈ 1.53 seconds.
For t = ( -96 - 45.25 ) / -32, we get t ≈ 5.53 seconds.
Therefore, the projectile will be at a height of 592 ft above the ground at approximately 1.53 seconds and again at 5.53 seconds.
b) Crash on the ground:
To find the time when the projectile will crash on the ground, we need to find the time when the height is zero. So we set h = 0 in the equation:
0 = -16t^2 + 96t + 480
Now, we can again solve this quadratic equation by factoring, completing the square, or using the quadratic formula. Using the quadratic formula:
t = ( -96 ± √(96^2 - 4(-16)(480)) ) / (2(-16))
Simplifying,
t = ( -96 ± √(9216 + 30720) ) / (-32)
t = ( -96 ± √39936 ) / (-32)
Now, let's calculate the square root of 39936:
√39936 = 199.84
So our equation becomes:
t = ( -96 ± 199.84 ) / -32
For t = ( -96 + 199.84 ) / -32, we get t ≈ -3.571 seconds.
For t = ( -96 - 199.84 ) / -32, we get t ≈ 9.571 seconds.
However, since time cannot be negative in this context, we discard the negative value and consider only the positive value.
Therefore, the projectile will crash on the ground at approximately 9.571 seconds.
So, the times when the projectile will be at a) a height of 592 ft above the ground and b) crash on the ground are approximately 1.53 seconds, 5.53 seconds, and 9.571 seconds.