A filmmaker wants to achieve an interesting visual effect by filming a scene through a converging lens with a focal length of 54.0 m. The lens is placed between the camera and a horse, which canters toward the camera at a constant speed of 7.8 m/s. The camera starts rolling when the horse is 41.0 m from the lens.
Find the average speed of the image of the horse:
(a) during the first 2.0 s after the camera starts rolling and
(b) during the following 2.0 s.
1/f=1/di+1/do
or
dido=fdo+fdi
taking the implicit derivative ( I hope you are taking calculus)
do(di/dt+ di(do/dt)=f do/dt+f di/dt
you are given di/dt (-7.8), given f, and you want di/dt when do=41
di/dt(do-f)=do/dt(f-di) . At this point you need di when do=41, again, use the lens equation for that di.
Solve for di/dt
for average speed, it's just distance_traveled/time-taken
To find the average speed of the image of the horse during each time interval, we need to calculate the distance traveled by the image and divide it by the time taken.
First, let's calculate the distance traveled by the image during the first 2.0 seconds after the camera starts rolling.
The distance traveled by the image can be found using the formula:
Distance = Speed × Time.
In this case, the speed of the image is the same as the speed of the horse, which is 7.8 m/s. The time is 2.0 seconds.
Distance (during first 2.0 s) = 7.8 m/s × 2.0 s = 15.6 m.
So, during the first 2.0 s after the camera starts rolling, the distance traveled by the image of the horse is 15.6 m.
Now, let's calculate the average speed during the first 2.0 s:
Average Speed (during first 2.0 s) = Distance / Time
= 15.6 m / 2.0 s
= 7.8 m/s.
Therefore, the average speed of the image of the horse during the first 2.0 s is 7.8 m/s.
Next, let's calculate the distance traveled by the image during the following 2.0 seconds.
The horse is cantering towards the camera, so the distance between the horse and the lens will decrease. The converging lens will form an image of the horse on the opposite side of the lens. The image formed by a converging lens is virtual, upright, and located on the same side as the object.
To calculate the distance traveled by the image of the horse during the next 2.0 seconds, we first need to find the position of the image after 2.0 seconds.
The position of the horse after 2.0 seconds can be calculated using the formula:
Position = Initial Position + Velocity × Time.
The initial position of the horse is 41.0 m from the lens, and the velocity of the horse is -7.8 m/s, as it is moving towards the lens (negative direction). The time is 2.0 seconds.
Position (after 2.0 s) = 41.0 m + (-7.8 m/s) × 2.0 s
= 41.0 m - 15.6 m
= 25.4 m.
So, after 2.0 seconds, the horse will be at a position of 25.4 m from the lens.
The distance traveled by the image during the next 2.0 seconds is given by:
Distance (during next 2.0 s) = Position (before 2.0 s) - Position (after 2.0 s)
= 41.0 m - 25.4 m
= 15.6 m.
Now that we have calculated the distance traveled by the image during the next 2.0 seconds, let's find the average speed:
Average Speed (during next 2.0 s) = Distance / Time
= 15.6 m / 2.0 s
= 7.8 m/s.
Therefore, the average speed of the image of the horse during the following 2.0 seconds is also 7.8 m/s.