Current Attempt in Progress
A woman on a bridge $97.5 \mathrm{~m}$ high sees a raft floating at a constant speed on the river below. She drops a stone from rest in an attempt to hit the raft. The stone is released when the raft has $8.19 \mathrm{~m}$ more to travel before passing under the bridge. The stone hits the water $4.57 \mathrm{~m}$ in front of the raft. Find the speed of the raft.
Number
Units

Step 1 ：First, we need to find the time it takes for the stone to fall using the equation for free fall: \(h = 0.5 * g * t^2\), where \(h\) is the height of the bridge, \(g\) is the acceleration due to gravity (9.81 m/s^2), and \(t\) is the time it takes for the stone to fall.

Step 2 ：Using the given height of the bridge (97.5 m), we can solve for \(t\): \(97.5 = 0.5 * 9.81 * t^2\)

Step 3 ：Solving for \(t\), we get \(t \approx 4.46\) seconds.

Step 4 ：Next, we need to find the speed of the raft using the equation: \(distance = speed * time\), where distance is the distance the raft travels before the stone hits the water (8.19 m + 4.57 m), speed is the speed of the raft, and time is the time it takes for the stone to fall.

Step 5 ：Using the given distance (12.76 m) and the time we found earlier (4.46 seconds), we can solve for the speed of the raft: \(12.76 = speed * 4.46\)

Step 6 ：Solving for the speed, we get \(speed \approx 2.86\) m/s.

Step 7 ：\(\boxed{\text{The speed of the raft is approximately 2.86 m/s}}\)