Please help! I'm having the worst time figuring this out.
3. One of the fireworks is launched from the top of the building that is 70ft tall with an initial
upward velocity of 150 ft/sec.
a. What is the equation for this situation?
b. When will the firework land if it does not explode? I think the firework will land in 3 seconds.
Well, who does feet any more but anyway in that system
g = -32 ft/s^2
v = Vi - g t
v = 150 - 32 t
at the top, v = 0 so
0 = 150 - 32 t
t = 4.69 seconds to the top
h = Hi + Vi t - (1/2)gt^2
h = Hi + Vi t - 16 t^2
h = 70 + 150 t - 16 t^2
so at the top
h = 70 + 150(4.69) -16 (4.69)^2
h = 421.5 feet high at the top
Now it has to fall from there
how long to fall from 421.5 ft
421.5 = (1/2)(32)t^2
t to fall = 5.13 seconds fall time
so total time in air =4.69+5.13
= 9.82 seconds
note I could have solved for total t directly by saying
0 = 70 + 150 t - 16 t^2
but I did not want to solve the quadratic equation so did upward and downward as separate problems.
Thank you! Do you know how I can make a table that shows the height from time t=0 until it hits the ground?
here is the height versus time t
h = 70 + 150 t - 16 t^2
Sure, I'd be happy to help you figure this out!
a. The equation that describes the motion of a projectile launched vertically upward is given by the formula:
h(t) = h₀ + v₀t - (1/2)gt²
Where:
h(t) is the height of the projectile at time t,
h₀ is the initial height (the top of the building, in this case),
v₀ is the initial velocity (positive since it's upward in this case),
t is the time elapsed, and
g is the acceleration due to gravity (which is approximately 32.2 ft/s²).
Using this equation, we can calculate the height of the firework at any given time.
b. To determine when the firework will land, you need to find the value of t when the height of the firework (h(t)) equals zero. This is because the firework will have returned to the ground when it reaches a height of zero.
So, you can set up the equation:
0 = h₀ + v₀t - (1/2)gt²
Substituting the given values:
h₀ = 70 ft (height of the building)
v₀ = 150 ft/s (initial upward velocity)
0 = 70 + 150t - (1/2)(32.2)t²
Now, you can solve this equation to find the values of t when the firework will land. In this case, it appears you already have an estimated value of 3 seconds. To verify if it's correct, we can substitute t = 3 into the equation and see if the height is indeed zero.
h(3) = 70 + 150(3) - (1/2)(32.2)(3)²
= 70 + 450 - (1/2)(32.2)(9)
= 70 + 450 - 145.35
= 374.65
Since the height at t = 3 seconds is not zero but rather 374.65 feet, it means the firework has not landed yet. Therefore, your initial estimate of 3 seconds is incorrect.
To find the exact time when the firework will land, you can rearrange the equation to solve for t. This can be done by setting the equation equal to zero (since that's when the firework lands) and then using the quadratic formula to solve for t.