Using the formula y=-16t^2+vt+h help me with this word problem : A firework rocket is fired straight up into the air with an initial velocity of v=192ft/sec
From the top of an apartment building h=48ft off the ground. The firework will explode when it reaches the peak (maximum) of its vertical trajectory. At what will the firework explode & with what initial velocity should the firework be fired to explode at the height of 948 feet?
The firework will explode at a height of 948 feet when it is fired with an initial velocity of v=1,056 ft/sec. To solve this problem, we can use the formula y=-16t^2+vt+h, where y is the height of the firework, t is the time it takes to reach the peak of its trajectory, v is the initial velocity, and h is the initial height.
We can rearrange the equation to solve for t: t = (v - sqrt(v^2 - 4(-16)(h-y)))/(2(-16)).
Plugging in the given values, we get t = (1,056 - sqrt(1,056^2 - 4(-16)(48-948)))/(2(-16)) = 4.5 seconds.
Therefore, the firework will explode at a height of 948 feet when it is fired with an initial velocity of 1,056 ft/sec.
To find the time at which the firework will explode, we need to find the time when the vertical displacement (y) is equal to the height at which it will explode. In this case, the height is 948 feet.
Given:
Initial velocity (v) = 192 ft/sec
Height from the ground (h) = 48 ft
Desired height at explosion (y) = 948 ft
Using the formula y = -16t^2 + vt + h, we can plug in the values and solve for time (t):
948 = -16t^2 + 192t + 48
Now, let's rearrange the equation to solve for t:
16t^2 - 192t + 48 - 948 = 0
16t^2 - 192t - 900 = 0
To make the equation easier to solve, let's divide through by 4:
4t^2 - 48t - 225 = 0
We can solve this quadratic equation by factoring or using the quadratic formula. In this case, we'll use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
For our equation 4t^2 - 48t - 225 = 0, a = 4, b = -48, and c = -225. Plugging these values into the quadratic formula, we get:
t = (-(-48) ± √((-48)^2 - 4 * 4 * (-225))) / (2 * 4)
t = (48 ± √(2304 + 3600)) / 8
t = (48 ± √5904) / 8
Now, we'll calculate the value inside the square root:
√5904 ≈ 76.81
Plugging this value back into the formula for t:
t = (48 ± 76.81) / 8
We have two possible solutions:
t1 = (48 + 76.81) / 8 ≈ 17.851
t2 = (48 - 76.81) / 8 ≈ -3.601
Since time cannot be negative in this context, we'll discard t2. Therefore, the firework will explode after approximately 17.851 seconds.
To find the initial velocity (v) required for the firework to explode at a height of 948 feet, we can use the same formula. This time, we'll set the height at explosion (y) to 948 feet and solve for the initial velocity.
Given:
Height from the ground (h) = 48 ft
Height at explosion (y) = 948 ft
Time (t) = 17.851 seconds
Using the formula y = -16t^2 + vt + h, we can now solve for v:
948 = -16 * (17.851)^2 + v * 17.851 + 48
Now, let's rearrange the equation to solve for v:
-16 * (17.851)^2 + v * 17.851 = 948 - 48
v * 17.851 = 900 + 948
v * 17.851 = 1848
v = 1848 / 17.851
v ≈ 103.45
Therefore, to explode at a height of 948 feet, the firework should be fired with an initial velocity of approximately 103.45 ft/sec.
To find when the firework will explode and the initial velocity required for it to explode at a height of 948 feet, we can use the formula y = -16t^2 + vt + h, where y is the height of the firework at time t, v is the initial velocity, and h is the initial height.
Let's start with the first part of the problem:
The firework is fired with an initial velocity of v = 192 ft/sec, and from the top of an apartment building at a height of h = 48 ft off the ground.
To find when the firework will explode, we need to determine when it reaches its peak, which is the maximum height of its trajectory.
At the peak, the vertical velocity of the firework is 0, meaning its upward motion stops and starts to fall back down.
We can find the time it takes for the firework to reach its peak by setting the vertical velocity to 0 in the formula.
0 = -16t^2 + 192t + 48
To solve for t, we need to factor or use the quadratic formula. In this case, let's use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
In our formula, a is -16, b is 192, and c is 48. Plugging these values into the quadratic formula, we get:
t = (-(192) ± √((192)^2 - 4(-16)(48))) / (2(-16))
Now we can simplify the equation:
t = (-192 ± √(36864 + 3072)) / (-32)
t = (-192 ± √39936) / (-32)
t = (-192 ± 199.84) / (-32)
Now, we have two possible solutions for t:
t = (-192 + 199.84) / (-32) or t = (-192 - 199.84) / (-32)
Simplifying further:
t = 7.84 / (-32) or t = -391.84 / (-32)
t = -0.245 or t = 12.244
We disregard the negative solution since time cannot be negative in this context.
Therefore, the firework will explode at approximately t = 12.244 seconds.
Now, let's move on to the second part of the problem:
We want to find the initial velocity required for the firework to explode at a height of 948 feet.
Using the same formula as before, we plug in the new height (y) and solve for v:
948 = -16t^2 + vt + 48
At the time the firework explodes, t = 12.244 (as calculated previously).
Plugging this into the equation:
948 = -16(12.244)^2 + v(12.244) + 48
Simplifying:
948 = -2988.724 + 12.244v + 48
We isolate v:
12.244v = 2988.724 - 960
12.244v = 2028.724
v = 165.444
Therefore, to explode at a height of 948 feet, the firework should be fired with an initial velocity of approximately 165.444 ft/sec.