At 10:00 .hrs a 1.5- m-long vertical stick in air casts a shadow 1.4 m long. If the same stick is placed at 10:00 hrs in air in a flat bottomed pool of salt water half the height of the stick, how long is the shadow on the floor of the pool? (For this pool, n = 1.58.) I solved for the inc-angle and got 43.025deg. Solved for ref-angle and got 25.58deg. Quite confused on what to do next.
the shadow's length is related to the angle of refraction and depth.
tanTheta2=length/depth
You know in air, the angle of incidence is given by arctanTHETA1=1.4/1.5
OR THETA1=43.025 degrees as you found.
Now, to the shadow. find angle of refraction...I wont check your, but assume 25.58
then length shadow= depth*tan25.58
and depth is .5 meters. So the length must be from that formula. HOWEVER, that is the length of shadow of the stick above the water. Sunlight also produces a shadown length for the part of the stick under water. If you draw your ray diagrams, you will convince yourself it is connected to the shadow base of the stick above the water, so the entire shadow should be
1m*tan25.58
To solve this problem, you have already found the incident angle and the refracted angle, which is a great start. Now, let's use these angles to determine the length of the shadow on the floor of the pool.
The incident angle, measured from the vertical axis to the incoming light ray, can be calculated using the formula:
sin(incident angle) = height of stick / length of shadow in air
Plugging in the values, we have:
sin(incident angle) = 1.5 m / 1.4 m
Using a calculator, you can find the value of the incident angle.
Now that you have the incident angle, you can calculate the refracted angle using Snell's law:
sin(refracted angle) = (n1 / n2) * sin(incident angle)
In this case, n1 is the refractive index of air (which is approximately 1) and n2 is the refractive index of salt water (given as 1.58). Plugging in these values along with the incident angle, you can solve for the refracted angle.
Once you have the refracted angle, you can use it to calculate the length of the shadow in the pool using trigonometry. Recall that the length of the shadow is equal to the tangent of the refracted angle multiplied by the depth of the pool:
length of shadow in pool = tan(refracted angle) * (height of stick / 2)
Since the height of the stick in the pool is half the actual length of the stick, we divide by 2.
Now you can calculate the length of the shadow on the floor of the pool by plugging in the values you have found.