Ironman steps from the top of a tall building. He falls freely from rest to the ground a distance of h. He falls a distance of h/ 3 in the last interval of time of 1.1 s of his fall.
Hint: First, compute the velocity when Ironman reaches the height equal to the distance fallen. This requires that you do the following: define origin as the bottom of the building. Then use x-x0 = -v0*(t-t0)-(1/2)g(t-t0)^2 where x=0 and x0= (distance fallen) and t-t0 is the time interval given. In this formulation, you are going to get magnitude of v0 since you already inserted the sign.
You then insert v0 that you just calculated into the kinematic equation that involves v, g, and displacement (v^2-v0^2 = 2g(height-(distance fallen)), but now v (which is the final velocity is v0 from above) and v0 in this case is the velocity that the Ironman has when he begins to fall, which is 0.
This gives a quadratic equation for height h, and you will need to use the binomial equation to solve for h. Choose the larger of the two solutions.
To solve the problem, we need to find the height (h) from which Ironman fell. We are given that Ironman falls a distance of h/3 in the last interval of time (t) of 1.1 s of his fall.
First, let's define the origin as the bottom of the building. We can use the kinematic equation for displacement to calculate the initial velocity (v0) when Ironman reaches the height equal to the distance fallen.
The kinematic equation for displacement is given by:
x - x0 = -v0 * (t - t0) - (1/2) * g * (t - t0)^2
Here, x is the final position (0, as Ironman reaches the ground), x0 is the initial position (the distance fallen), v0 is the initial velocity, t is the final time (t = 1.1 s), t0 is the initial time (t0 = 0), and g is the acceleration due to gravity.
Since Ironman falls from rest, the initial velocity (v0) is 0. Therefore, the equation becomes:
-x0 = -(1/2) * g * t^2
Now, we can solve for the magnitude of the initial velocity (|v0|):
|v0| = sqrt(2 * g * x0 / t^2)
Next, we insert the calculated |v0| into the kinematic equation that involves final velocity (v), acceleration due to gravity (g), and displacement (height - distance fallen):
v^2 - v0^2 = 2 * g * (height - x0)
In this case, v is the final velocity, which is the same as the magnitude of the initial velocity (|v0|) since Ironman falls from rest.
Therefore, the equation becomes:
(|v0|)^2 - 0 = 2 * g * (height - x0)
Simplifying further:
2 * g * (height - x0) = g * t^2
Canceling out the gravitational acceleration (g):
2 * (height - x0) = t^2
Finally, we can solve the quadratic equation for height (h) using the binomial formula. The equation is:
2 * h - 2 * x0 = t^2
2 * h = t^2 + 2 * x0
h = (t^2 + 2 * x0) / 2
Now, substitute the given values and solve for h:
h = (1.1^2 + 2 * (h/3)) / 2
Multiply the equation by 2 (to remove the denominator):
2h = 1.21 + (2/3)h
Multiply both sides by 3 (to remove fractions):
6h = 3.63 + 2h
Subtract 2h from both sides:
4h = 3.63
Divide both sides by 4:
h = 0.9075
Therefore, the height from which Ironman fell is approximately 0.9075 meters.