You are the pilot of an airplane sent to drop supplies to victims of an accident that are stranded on a small island. The plane's altitude is always 500 meters and your speed is 89.61 m/s.

1. Once released, how much time will elapse before the supply package reaches the level of the island?

   (Hint: Use the distance equation in the y-direction.) 
   Step 1: d_i = 500 m; a = -9.8 m/s² (gravity, negative sign indicates downward direction); d_f = 0 m; v_i = 0 m/s
   where d_i = initial distance and d_f = final distance
   Step 2: Solve the equation d_f = ( 1 / 2 ) at² + v_it + d_i for t. So,

   
2. Will the descent time of the supply package change if the airplane's speed changes?

   (a) Yes (b) No

    

3. At what (horizontal) distance in front of the island should the package be released in order to hit the island?
   Step 1: During descent the package moves forward a horizontal distance of d_x.

   v = 89.61 m/s; t = 10.10 sec

   Step 2: d_x = vt = 905.06 m. The airplane must release the package at least 905.06 m in front of the island.

4. What is the package's horizontal speed when it reaches the level of the island? 

   Neglecting air resistance, it remains constant. v_x = 89.61 m/s

    

5. What is the package's vertical speed when it reaches the level of the island?
   Step 1: v_i = 0 m/s; a = -9.8 m/s²; t = 10.10 sec

   Step 2: Solve the equation a = (v_f - v_i )/t for v_f. So, v_f = at + v_i = -98.98 m/s ( v_f = v_y )

6. What will be the package's flight angle with respect to the level of the island as it descends?
   Step 1: Draw the diagram,

   Step 2: v_y = -98.98 m/s; v_x = 89.61 m/s
   Step 3: tan Q = v_y / v_x; Q = tan^-1(v_y / v_x) = 47.84 ^o

   Suppose the victims on the island can retrieve supply packages that land within 30 meters of the island. The length of the island is 50 meters along the direction you are approaching.

    

7. How far in front of the island would the airplane have to release the supply package for it to land 30 meters in front of the island?

   d_x(front) = distance from problem #3 + 30 m = 935.06 m

    

8. How far in front of the island would the airplane have to release the supply package for it to land 30 meters in back of the island?

   d_x(back) = distance from problem #3 - 50 m - 30 m = 825.06 m

    

9. Time₁ will be the time on your watch when you release the package and it lands 30 meters in front of the island. Time₂ will be the time on your watch when you release the package and it lands 30 meters in back of the island. Calculate the amount of time you have to successfully drop the package, namely, time₁- time₂.
   Step 1: time₁ = 0 sec; v_x = 89.61 m/s

   Step 2: Solve the equation d_x = (v_x )(time₂) for time₂. Also realize that d_x = d_x(front) - d_x(back).

   time₂ = [ d_x(front) - d_x(back) ] / v_x = 1.23 sec. Therefore, time₁ - time₂ = 1.23 sec

10. What could be done to decrease the package's speed when it reaches the ground?
    Use a parachute.