Question

You have a battleship sitting in a fjord in the coast of Iceland. There is a volcanic mountain peak on which an alien spacecraft has landed. With your
adjustable-velocity 10-inch rifle, you have to take them out. The only thing is: the adjustable velocity rifle can only fire at a 45º angle above horizontal.
i) Your battleship (B) is 5.000 km from shore point S1.
ii) Shore point S1 is also the base of the volcano, which is perfectly symmetric.
iii) On the far side of the volcano is shore point S2, where the volcano returns to sea level.
iv) The distance from S1 to S2 is 30.00 km; the peak is directly above the midpoint of that base, and it is 3.000 km above sea level.
v) The aliens (A) are at elevation 2250 m, on the far side of the volcano.
TASKS:
(a) Analyze your first shot: it leaves the adjustable velocity rifle at standard speed 500 m/sec, at an angle 45º above horizontal.
(1) Derive: the formula for x(t) and y(t) for this first artillery shot. Write them down neatly, with all numeric values specified.
(2) Decide: does this round hit the aliens at point A? Or, does it land uphill from them? Downhill? (Circle your answer.)
(3) Adjust if necessary: How would you adjust the velocity to guarantee a perfect hit? More speed or less speed? (Circle your answer.)
(b) Calculate the perfect launch velocity at 45º launch angle for absolute smackdown direct hit on the alien spacecraft. Show your work neatly.

Answer 1

1) The initial velocity, u = 500 m/s

We have to convert it into the x and y components.

The y component is

The displacement in the y direction is given by

Taking g = 9.8 m/s^2,

(ans 1)

The displacement in x direction is

(ans 2)

2) For a hit to happen, the horizontal distance the projectile must reach the point where h = 2250 m

Taking h(S2) = 0

and the height at the topmost point = 3000 m,

The topmost point is at d = 5000 + 30000/2 = 20000 m

S2 is at 35000 m.

So, the distance between the peak and S2 is 15000 m

With this we can get

This is the distance from S2.

The distance from the ship is

x = 35000-11250 = 23750 m

Using this as x(t) we get

The vertical position of the projectile is

The actual height it should have reached is 2250 m.

Since the height it has reached is lower than the actual height, the projectile lands uphill from the alien spaceship.

3) To make the projectile to move further downhill, to the aliens, the velocity must be increased.

b) The point at which the hit must happen is x = 23750 and y = 2250.

Using these in the equation,

Since the launch angle is 45 degrees, and

,

Subtracting the first equation from the second,

Using the second equation,

Remember that this v is the x or y component of the velocity and not the launch velocity