A concave mirror has a focal length f = 200 [cm]. The mirror is lying flat on the floor. You take a ball and align it with the center (vertex point) of the mirror. Then at time (t=0) you release the ball from a height h = 5 [m].
(a) The ball falls and hits the center of the mirror. Describe the height of the image di = zi as a function of time from the time it is released until it hits the mirror. In other word find how the height of the image changes as function of time. (b) At what time the ball goes through the focal point of the mirror
To find the height of the image (di) as a function of time, we can use the mirror equation and the magnification equation for concave mirrors. Let's break down the problem step by step.
(a) Finding the height of the image as a function of time (di):
1. We know that the focal length of the mirror (f) is 200 cm. Since the mirror is concave, the focal length is positive.
2. The mirror is lying flat on the floor, so the vertex of the mirror is at a height of 0 cm.
3. At time (t=0), the ball is released from a height (h) of 5 m. We need to convert this height into centimeters for consistency with the focal length.
h = 5 m = 500 cm.
Now let's calculate the height of the image at any given time (t).
4. Recall the mirror equation: 1/f = 1/v + 1/u
Since the mirror is lying flat, the object distance (u) is negative and equal to the height (h) of the ball above the mirror surface: u = -h.
The image distance (v) is positive and equal to the height of the image (di): v = di.
Substituting these values into the mirror equation, we have: 1/f = 1/di - 1/h.
5. Plug in the known values: f = 200 cm, h = 500 cm.
The mirror equation becomes: 1/200 = 1/di - 1/500.
6. Solve for the height of the image (di):
Rearrange the equation to isolate 1/di: 1/di = 1/200 + 1/500.
Combine the fractions: 1/di = (5 + 2)/1000 = 7/1000.
Take the reciprocal of both sides: di = 1000/7 cm.
Therefore, the height of the image (di) as a function of time is given by di = 1000/7 cm.
(b) To determine at what time the ball goes through the focal point of the mirror, we need to calculate the time it takes for the image to reach the focal point.
1. Recall that the focal point is the point where parallel rays of light converge or diverge after reflecting off the mirror.
2. For concave mirrors, when the object is placed at the focal point (u = f), the image formed is at infinity (v → ∞).
3. In this case, we want to find when the image is at the focal point (di = f).
Using our previously derived equation for the height of the image as a function of time:
di = 1000/7 cm = f.
4. Set di equal to f and solve for t (time):
1000/7 = 200.
1000t 7
------------- = ---.
7t - 500 1
Cross-multiply: 1000(7t - 500) = 7(1000t).
Expand: 7000t - 500000 = 7000t.
Combine like terms: -500000 = 0.
Since this equation leads to an invalid condition, there is no specific time at which the ball goes through the focal point of the mirror.