A stone is dropped into a well with initial velocity zero from a point 5 m above the mouth of the well and the splash is heard after 7 s. A second stone is thrown into the well with initial velocity uₒ from the same point and the splash is heard after 5 s.
Find the value of uₒ The velocity of sound is 350m/s.
To find the value of uₒ, we can use the equation of motion for the vertical motion of the stone.
The equation of motion for the stone can be written as:
s = ut + (1/2)at^2
Where:
s is the displacement (change in position)
u is the initial velocity
t is the time
a is the acceleration (in this case, it is the acceleration due to gravity, which is approximately 9.8 m/s²)
For the first stone:
s₁ = -5 m (negative because it is going downwards)
u₁ = 0 m/s (initial velocity is zero)
t₁ = 7 s
Substituting these values into the equation, we get:
-5 = 0 * 7 + (1/2) * 9.8 * 7^2
-5 = 24.5 * 7^2
-5 = 24.5 * 49
-5 = 1200.5
This is not a valid solution, so we move on to the second stone.
For the second stone:
s₂ = -5 m (negative because it is going downwards)
u₂ = uₒ m/s (initial velocity is uₒ)
t₂ = 5 s
Substituting these values into the equation, we get:
-5 = uₒ * 5 + (1/2) * 9.8 * 5^2
-5 = 5uₒ + 24.5 * 5^2
-5 = 5uₒ + 24.5 * 25
-5 = 5uₒ + 612.5
Rearranging the equation, we get:
5uₒ = -5 - 612.5
5uₒ = -617.5
uₒ = -617.5 / 5
uₒ = -123.5 m/s
Since the velocity cannot be negative, the value of uₒ is 123.5 m/s.
However, it's important to note that the negative sign in the equation represents the direction of the velocity. In this case, it means the stone was thrown downwards. So, if we consider the positive direction as upwards, the value of uₒ would be positive 123.5 m/s.