Question

A mountain climber stands at the top of a 50 meter cliff hanging over a calm pool of water. The climber throws two stone vertically 1 second apart and observes that they cause a single splash when they hit the water. The first stone has a initial velocity of +2.0 meters per second. What will the velocity of each stone be at the instant bitg stones hit the water?

Answer 1

One of two things is true about this question: EITHER it can't happen
as you've described it, OR you've left out some vital information.

-- IF the first stone was thrown downward with an initial speed and the
second one was dropped from rest 1 second later, then the second one
can never catch up with the first one, and they can never hit the water together.

-- IF the first stone was thrown downward with an initial speed, AND the
second one was released 1 second later, AND they actually do hit the
water together, THEN the second stone must have been given an initial
downward speed greater than 2 m/s, otherwise it could never catch up
with the first one.

Note:
The masses and weights of the stones are irrelevant and not needed.

=======================================================

An afterthought . . . . .

If the first stone was tossed UP at 2 m/s . . . that could be the meaning of the
prominent plus-sign that you wrote next to the 2 . . . then it rises for (2/9.8) second, then begins to fall, and passes the mountain climber's hand on the way down (4/9.8) second after he tossed it, falling at the same 2.0 m/s downward.

From there, it still has 50m to go before it hits the water.
50 = 2 T + 1/2 G T²
4.9 T² + 2 T - 50 = 0
T = 3 seconds

The first stone hits the water 3 seconds after passing the mountain climber's hand on the way down at a downward speed of 2.0 m/s. In that 3 seconds, it gains (3 x 9.8) = 29.4 m/s of additional speed, hitting the water at (29.4 + 2) = 31.4 m/s .

This is all just a guess, assuming that the 2.0 m/s was an UPWARD launch.
Maybe I'll come back later and calculate the second stone.

Answer 2

a) First stone, Vf₁ = 31.38 m/s

b) Second stone, Vf₂ = 34.81 m/s

Step-by-step explanation:

1) Data:

First stone:

Vo₁ = + 2 m/s (positive sign is understood as throw in the same direction of the acceleration, i.e. downward).

h = 50 m

t₁ = t

Vf₁ = ?

Second stone:

t ₂ = t - 1

Vf₂ = ?

2) Solution

i) First stone, final velocity:

Vf² = Vo² + 2gh

Vf² = (2 m/s)² + 2(9.81m/s²)(50m) = 984 m²/s²

Vf = 31.38 m/s

ii) First stone time, t

Vf = Vo + gt ⇒ t = [Vf - Vo] / g = [31.38 m/s - 2m/s] / 9.81 m/s² = 3.0 s

iii) Second stone time, t - 1

t₂ = t - 1 = 3.0s - 1.0s = 2.0s

iv) Second stone initial velocity, Vo₂

h = Vo₂ t + gt₂² / 2 ⇒ 50m = Vo₂ (2.0s) + 9.81m/s² (2.0s)² / 2

Vo₂ = [ 50m - 19.62m] / (2.0s) = 15.19 m/s

v) Second stone final velocity, Vf₂

Vf₂ = Vo₂ + gt₂ = 15.19 m/s + 9.81 m/s² (2.0s) = 34.81 m/s