A sled of mass 5.98 kg is coasting along on a frictionless ice-covered lake at a constant speed of 1.42 m/s. A 1.98-kg book is dropped vertically onto the sled. At what speed does the sled move once the book is on it?
To find the final speed of the sled once the book is dropped onto it, we need to apply the principle of conservation of momentum.
Conservation of momentum states that the total momentum before an event is equal to the total momentum after the event, provided that no external forces act on the system.
Let's denote the initial speed of the sled as V1 and the final speed of the sled (once the book is on it) as V2.
Since the sled is initially coasting along at a constant speed of 1.42 m/s and there are no external forces acting on it, its initial momentum is given by:
Momentum1 = mass of sled * initial speed of sled
= 5.98 kg * 1.42 m/s
When the book is dropped onto the sled, it adds to the combined mass of the system. So, the total mass of the sled and the book is the mass of the sled plus the mass of the book:
Total mass = mass of sled + mass of book
= 5.98 kg + 1.98 kg
To find the final speed of the sled (V2), we can use the conservation of momentum equation:
Momentum1 = Momentum2
(mass of sled * initial speed of sled) = (total mass * final speed of sled)
Substituting the given values into the equation:
(5.98 kg * 1.42 m/s) = ((5.98 kg + 1.98 kg) * V2)
Now, we can solve for V2:
(8.96 kg * 1.42 m/s) = (7.96 kg * V2)
12.7392 kg·m/s = 7.96 kg * V2
V2 = 12.7392 kg·m/s / 7.96 kg
V2 = 1.60 m/s
Therefore, the final speed of the sled, once the book is dropped onto it, is 1.60 m/s.