If an object is launched straight up into the air from a starting height of h_{0} feet, then the height of the object after t seconds is approximately h=-16t^2+v_{0}t+h_{0} feet, where v_{0} is the initial velocity of the object. Find the starting height and initial velocity of an object that attains a maximum height of 412 feet five seconds after being launched.
V = Vo + gt = 0.
Vo - 32*5 = 0.
Vo = 160 Ft/s.
h = ho + Vo*t + 0.5g*t^2 = 412 Ft.
ho + 160*5 - 16*5^2 = 412.
ho + 800 - 400 = 412.
ho + 400 = 412.
ho = 412 - 400 = 12 Ft.
2. A bullet is fired from the ground at an angle of 45o above the horizontal. What initial speed vo must the bullet have in order to hit a point 550 ft high on a tower located 600 ft away (ignoring air resistance)?
To find the starting height and initial velocity of the object, we need to use the given information and the equation h = -16t^2 + v_0t + h_0.
First, let's analyze the information given in the problem. It states that the object attains a maximum height of 412 feet after five seconds. This means that at some point, the object reaches its highest point in the air before falling back down.
When the object reaches its highest point, its vertical velocity is zero because it momentarily stops moving upwards before starting to fall back down due to the force of gravity. This means that after five seconds, the object's height formula can be simplified as follows:
h = -16t^2 + v_0t + h_0
At the maximum height, t = 5 and v = 0:
412 = -16(5)^2 + 0(5) + h_0
412 = -400 + h_0
To determine the starting height (h_0), we rearrange the equation:
h_0 = 412 + 400
h_0 = 812 feet
Therefore, the starting height of the object is 812 feet.
Now, to find the initial velocity (v_0), we note that the object reaches its maximum height after five seconds. To find the initial velocity, we need to find the velocity of the object at t = 0.
Using the equation for velocity, we can differentiate the equation for height with respect to time:
v = -32t + v_0
At t = 0, we can substitute the values to solve for v_0:
0 = -32(0) + v_0
0 = v_0
Therefore, the initial velocity of the object (v_0) is 0 feet per second.
In conclusion, the starting height of the object is 812 feet and the initial velocity is 0 feet per second.