Boulder rolling off of cliff: will it hit the village?

[Entr0py]

Homework Statement

A large boulder rests on a cliff 400 m above a small village in such a position that if it should roll off, it would leave with a speed of 50 m/s at an angle of 30 degrees below the horizontal. There is a pond with a diameter of 200 m, with its edge 100 m from the base of the cliff. The village houses are at the edge of the pond. A physics student says that the boulder will land in the pond. Is he right?

Homework Equations

y= y0 + v0yt + 1/2gt^2

x=R=v0xt

The Attempt at a Solution

Firstly, I need to find the vector components of the velocity. Since theta is -30 degrees and the magnitude of the velocity is 50 m/s, I find that v0x=50cos(30) which is 43.3 m/s and v0y=-50sin(30) which is -25 m/s.

Next I want to find out how long it would take the boulder to fall a vertical distance of 400 m (in this case, -400 m since i chose my y-axis to be positive in upwards direction). I use the equation y= y0 + v0yt + 1/2gt^2. I take y0 to be 0 m and y to be -400. Using the quadratic formula, I find that t=6.84 s.

Then I want to find the range or the horizontal distance the boulder travels, which I find using the equation x=v0xt. Since v0x=43.3 m/s and t=6.84 s, I plug those into get x=296 m.

I say that the physics student is right, that the boulder will land in the pond.

 

[haruspex]

How do you get a launch angle of 30 degrees?

 

[Entr0py]

From my texbook. The author puts in the diagram that the angle below the horizontal is 30 degrees.

 

[Entr0py]

Would that mean that v0y=-(50sin(30) or -25 m/s?

 

[haruspex]

Would that mean that v0y=-(50sin(30) or -25 m/s?

Yes. I was just about to point that out.

 

[Entr0py]

Then if I plug that in the quadratic formula, t=6.84 s. Now if I plug that into eqn. to find range I get 296.2 m. After all the physics student was right and the boulder WOULD NOT hit the village.

 

[haruspex]

Then if I plug that in the quadratic formula, t=6.84 s. Now if I plug that into eqn. to find range I get 296.2 m. After all the physics student was right and the boulder WOULD NOT hit the village.

Looks right to me.

 

[Entr0py]

Yes! I was looking at this question last night, thinking "I can't wait to solve this!" Next time, I'll be sure to look at the launch angle very carefully. Thanks for the response.

 

[Entr0py]

Homework Statement

A large boulder rests on a cliff 400 m above a small village in such a position that if it should roll off, it would leave with a speed of 50 m/s. There is a pond with a diameter of 200 m, with its edge 100 m from the base of the cliff. The village houses are at the edge of the pond. A physics student says that the boulder will land in the pond. Is he right?

Homework Equations

y= y0 + v0yt + 1/2gt^2

x=R=v0xt

The Attempt at a Solution

Firstly, I need to find the vector components of the velocity. Since theta is 30 degrees and the magnitude of the velocity is 50 m/s, I find that v0x=50cos(30) which is 43.3 m/s and v0y=50sin(30) which is 25 m/s.

Next I want to find out how long it would take the boulder to fall a vertical distance of 400 m (in this case, -400 m since i chose my y-axis to be positive in upwards direction). I use the equation y= y0 + v0yt + 1/2gt^2. I take y0 to be 0 m and y to be -400. Using the quadratic formula, I find that t=11.9 seconds.

Then I want to find the range or the horizontal distance the boulder travels, which I find using the equation x=v0xt. Since v0x=43.3 m/s and t=11.9 s, I plug those into get x=515.3 m.

I say that the physics student is wrong, that the boulder won't land in the pond 300 m away from the mountain, but will instead hit the village houses.

EDIT: the physics student is right.

 

[haruspex]

Yes! I was looking at this question last night, thinking "I can't wait to solve this!" Next time, I'll be sure to look at the launch angle very carefully. Thanks for the response.

... and perhaps mention it in the first post? :-)

 

[Entr0py]

Yes, I will. I just looked at the diagram and said "Look at that 30 degrees!" but didn't notice that it was BELOW the horizontal until I rechecked.

 

[haruspex]

Yes, I will. I just looked at the diagram and said "Look at that 30 degrees!" but didn't notice that it was BELOW the horizontal until I rechecked.

Had it been above the horizontal it no doubt would have stayed safely in place on the cliff.

 

[Entr0py]

Had it been above the horizontal it no doubt would have stayed safely in place on the cliff.

That didn't even cross my mind. I need a common sense pill.

 

[Entr0py]

Now to find how fast the boulder is going when it hits the ground I would use vf^2=v0^2 + 2a(x-x0). Which would mean that vf^2=50^2 - (19.6*-400) or vf=-101.7 m/s?

 

[Entr0py]

Also, wouldn't its horizontal component of velocity just before it hits be found by v(x)= v0x +a(x)t? But since there is no horizontal acceleration, then it would be v(x)=v0x, since gravity does not act horizontally on the boulder, which would make v(x)=50cos(-30) or 43.3 m/s.

 

[haruspex]

Now to find how fast the boulder is going when it hits the ground I would use vf^2=v0^2 + 2a(x-x0). Which would mean that vf^2=50^2 - (19.6*-400) or vf=-101.7 m/s?

Yes.

Also, wouldn't its horizontal component of velocity just before it hits be found by v(x)= v0x +a(x)t? But since there is no horizontal acceleration, then it would be v(x)=v0x, since gravity does not act horizontally on the boulder, which would make v(x)=50cos(-30) or 43.3 m/s.

Yes.

 

[Entr0py]

Gracias