An interesting problem, and thanks to the precise heading you put for the question.
We will assume zero air resistance. We further assume that the angle with vertical is t=53.13 degrees, corresponding to sin(t)=0.8, and therefore cos(t)=0.6.
Given: angle with vertical, t = 53.13 degrees sin(t)=0.8; cos(t)=0.6; air-borne time, T = 20 seconds initial height, y0 = 800 m
Assume g = -9.81 m/s^2
initial velocity, v m/s (to be determined)
Solution:
(i) Determine initial velocity, v. initial vertical velocity, vy = vsin(t)=0.8v Using kinematics equation, S(T)=800+(vy)T+(1/2)aT^2 ....(1) Where S is height measured from ground.
substitute values in (1): S(20)=800+(0.8v)T+(-9.81)T^2 => v=((1/2)9.81(20^2)-800)/(0.8(20))=72.625 m/s for T=20 s
(ii) maximum height attained by the bomb Differentiate (1) with respect to T, and equate to zero to find maximum dS/dt=(vy)+aT=0 => Tmax=-(vy)/a = -0.8*72.625/(-9.81)= 5.9225 s
Maximum height, Smax =S(5.9225) =800+(0.8*122.625)*(5.9225)+(1/2)(-9.81)(5.9225^2) = 972.0494 m
(iii) Horizontal distance travelled by the bomb while air-borne Horizontal velocity = vx = vcos(t) = 0.6v = 43.575 m/s Horizontal distace travelled, Sx = (vx)T = 43.575*20 = 871.5 m
(iv) Velocity of the bomb when it strikes ground vertical velocity with respect to time V(T) =vy+aT...................(2) Substitute values, vy=58.1 m/s, a=-9.81 m/s^2 V(T) = 58.130 + (-9.81)T => V(20)=58.130-(9.81)(20) = -138.1 m/s (vertical velocity at strike)
vx = 43.575 m/s (horizontal at strike) resultant velocity = sqrt(43.575^2+(-138.1)^2) = 144.812 m/s (magnitude) in direction theta = atan(43.575,138.1) = 17.5 degrees with the vertical, downward and forward. (direction)
Your friend moved along the shore due to ; The swash effect and the Backwash effect
Swash effect is caused by the upsurge of water up along the slopping front of the beach and this same upsurge in water moves back into the beach in what is known as the backwash effect hence the movement of your friend form where they were in the surf zone to another position still within the surface zone is caused by the BACKWASH EFFECT