How Fast Does the Shadow Move on a Dome as a Balloon Rises?

[Myung]

Homework Statement

A hemispherical dome has a diameter of 100m. A search light was placed at point A as shown at the middle of the dome at B. A balloon was released vertically at a velocity of 4m/s. How fast will the shadow of the balloon move alone the roof if it traveled 25 m vertically.

Image link:http://bit.ly/rem8PG

Homework Equations

The Attempt at a Solution

I don't know how to start this. I don't understand the nature of the shadow and why did it bounce off the wall like as shown in the image.

What relationship should i use? Please help me to solve this step by step. I'm willing to learn.

 

[Mark44]

Homework Statement

A hemispherical dome has a diameter of 100m. A search light was placed at point A as shown at the middle of the dome at B. A balloon was released vertically at a velocity of 4m/s. How fast will the shadow of the balloon move alone the roof if it traveled 25 m vertically.

Image link:http://bit.ly/rem8PG

Homework Equations

The Attempt at a Solution

I don't know how to start this. I don't understand the nature of the shadow and why did it bounce off the wall like as shown in the image.

What relationship should i use? Please help me to solve this step by step. I'm willing to learn.

The shadow isn't bouncing off the inside of the dome. That line is just a radius of the dome, from the light out to the hemispherical wall.

 

[Myung]

The shadow isn't bouncing off the inside of the dome. That line is just a radius of the dome, from the light out to the hemispherical wall.

Oh, so using that the value of the shadow will have height of 25 m which is 50m from point B after the instant of traveling upwards. But what is the use of the velocity of the balloon? What variables can I use to get the speed of the shadow?

 

[Mark44]

The shadow travels along the curved surface of the dome. As the balloon rised 25 m. the shadow will move along an arc whose length is r * $\theta]$.

 

[Myung]

The shadow travels along the curved surface of the dome. As the balloon rised 25 m. the shadow will move along an arc whose length is r * $\theta]$.

I can't get θ without knowing how long did the balloon fly over B. If I only how long it took then I would have a fixed value after it traveled solving an angle at point A and getting θ by sine-law and supplementary angles.

 

[Myung]

This is what I have so far:

Since we are looking for velocity of shadow
let that be = d(h)/d(t)

and height of shadow = h

So Sinθ = h/50m
derivative of this equation is:
Cosθ(dθ/dt) = [50m(1)[d(h)/d(t)] - h(0)] / 2500



I have no values for θ and (dθ/dt) if I can get this I can get the d(h)/d(t) which is the main requirement of this equation. Still I haven't used the 4m/s velocity of the balloon from point B, I know that has a significant role in solving this but I can't seem to use it. I tried to get it's height after the instant but there is no time given. Help guys?

 

[Myung]

The shadow travels along the curved surface of the dome. As the balloon rised 25 m. the shadow will move along an arc whose length is r * $\theta]$.

Thank you this helped me solve the equation. The speed of the shadow is 6.4m/s