A stone is dropped into a river from a bridge 43.9 m above the water. Another stone is dropped vertically down 1s after the first is dropped. Both stones strike the water at the same time.
a) What is the initial speed of the second stone?
Write an equation for the vertical height of each stone. Let t=0 be the time the first stone is dropped. The second stone must have an initial vertical (downward) velocity so it can catch up with the first.
Y1 = 43.9 - (g/2) t^2
Y2 = 43.9 - Vo (t-1) - (g/2)(t-1)^2
Set Y1 = 0 and Y2 = 0 (the heights when each hit the ground). Use the first equation to solve for t. Use the second equation to solve for Vo, the initial downward velocity of the second stone. Note that t-1 is the time of flight of the second stone.
To solve for the initial speed of the second stone, we can follow the steps provided in the problem.
Step 1: Write the equation for the vertical height of each stone.
The equation for the height of an object in free fall is given by:
Y = Y0 + V0t - (1/2)gt^2
For the first stone (let's call it Stone 1), dropped at t=0, the equation becomes:
Y1 = 43.9 - (1/2)gt^2
For the second stone (let's call it Stone 2), dropped 1 second after the first stone, the equation becomes:
Y2 = 43.9 - Vo(t-1) - (1/2)g(t-1)^2
Where:
- Y1 and Y2 are the respective heights of the stones.
- g is the acceleration due to gravity (approximately 9.8 m/s^2).
- Vo is the initial downward velocity of the second stone.
- t-1 is the time of flight of the second stone (since it was dropped 1 second after the first stone).
Step 2: Set Y1 = 0 and Y2 = 0 (the heights when each stone hits the ground).
Since both stones strike the water at the same time, we set both equations equal to zero:
43.9 - (1/2)gt^2 = 0
43.9 - Vo(t-1) - (1/2)g(t-1)^2 = 0
Step 3: Solve for t, the time it takes for both stones to hit the water.
For the first equation, rearrange to solve for t:
(1/2)gt^2 = 43.9
gt^2 = 87.8
t^2 = 87.8/g
t = √(87.8/g)
Step 4: Solve for Vo, the initial downward velocity of the second stone.
Substitute the value of t found in Step 3 into the second equation:
43.9 - Vo(√(87.8/g) - 1) - (1/2)g(√(87.8/g) - 1)^2 = 0
Simplify and isolate Vo:
Vo(√(87.8/g) - 1) = 43.9 - (1/2)g(√(87.8/g) - 1)^2
Vo = (43.9 - (1/2)g(√(87.8/g) - 1)^2) / (√(87.8/g) - 1)
That's the equation to solve for Vo, the initial speed of the second stone. You can substitute the value of g (9.8 m/s^2) to find the specific numerical result.