math professor is thrown off the matthews bridges, height 240ft, by a disgruntled calculus class with an intitial upward velocity of 10ft/s
a) assumign the acceleration dude to gravity is -32ft/s^2,find the macimum height of the professor.
b) find the time before the professor is wimming with the fishes
To find the maximum height of the professor, we can use the kinematic equation for vertical motion:
h = h₀ + v₀t + (1/2)at²
where:
h = height
h₀ = initial height (240 ft)
v₀ = initial velocity (10 ft/s)
a = acceleration due to gravity (-32 ft/s²)
t = time
a) Maximum Height:
To find the maximum height, we can determine the time it takes for the professor to reach the topmost point, where the velocity becomes zero (v = 0). At this point, the professor starts falling downward. So, we set v = 0 and solve for t:
v = v₀ + at
0 = 10 - 32t
32t = 10
t = 10/32 = 0.3125 seconds (approx)
Now, we can substitute the found time (t) back into the kinematic equation to find the maximum height (h):
h = h₀ + v₀t + (1/2)at²
h = 240 + 10(0.3125) + (1/2)(-32)(0.3125)²
h ≈ 240 + 3.125 - 1 (approx)
h ≈ 242.125 ft
Therefore, the maximum height of the professor is approximately 242.125 ft.
b) Time before professor is underwater:
To find the time before the professor falls into the water, we can use the kinematic equation for vertical motion:
h = h₀ + v₀t + (1/2)at²
In this case, we want to find the time when the height (h) is equal to zero (h = 0). So we set h = 0 and solve for t:
0 = 240 + 10t + (1/2)(-32)t²
This equation is a quadratic equation, and we can solve it using various methods like factoring or the quadratic formula. Here, we will use the quadratic formula:
t = (-b ± √(b² - 4ac)) / (2a)
For this equation, a = (1/2)(-32) = -16, b = 10, and c = 240. Now, let's substitute these values into the quadratic formula:
t = (-10 ± √(10² - 4(-16)(240))) / (2(-16))
t = (-10 ± √(100 + 15360)) / (-32)
t = (-10 ± √15460) / (-32)
Simplifying this further and solving for t will give us the two possible times at which the professor reaches the water level.
Note: In this case, we will only consider the positive value, as the negative value has no physical significance in this context.
Therefore, the time before the professor is "swimming with the fishes" is the positive value of t obtained from the quadratic formula.