Two stones of different masses are thrown straight upward one on earth and on the moon (where gravity produces an acceleration of 1.67m/second squared) Both reach the same height. if the stone on the moon has an initial speed v, what was the initial speed of the stone on earth in terms of v?
To solve this problem, we need to compare the motion of the stones on Earth and on the Moon.
We know that the stones reach the same height, which implies that the maximum height is achieved when the final velocity becomes zero.
Let's denote the initial velocity of the stone on Earth as V_(E) and the initial velocity of the stone on the Moon as V_(M). We need to find the relation between V_(E) and V_(M) in terms of v, the initial speed of the stone on the Moon.
We can start by using the kinematic equation for velocity:
V^2 = u^2 + 2as
Where:
V is the final velocity,
u is the initial velocity,
a is the acceleration,
and s is the displacement.
For both stones, the displacement s is the same because they both reach the same height. The acceleration on Earth is approximately 9.8 m/s², while on the Moon it is 1.67 m/s².
For the stone on Earth:
V_(E) = 0 (final velocity is zero when it reaches maximum height)
u_(E) = ?
a_(E) = 9.8 m/s²
For the stone on the Moon:
V_(M) = 0 (final velocity is zero when it reaches maximum height)
u_(M) = v (initial speed)
a_(M) = 1.67 m/s²
Using the equation for the stone on Earth, we have:
0 = u_(E)^2 + 2 * 9.8 * s
Using the equation for the stone on the Moon, we have:
0 = v^2 + 2 * 1.67 * s
Since both stones reach the same height, the displacement s is the same in both equations.
By equating the two equations, we can solve for u_(E) in terms of v:
u_(E)^2 + 2 * 9.8 * s = v^2 + 2 * 1.67 * s
Simplifying the equation:
u_(E)^2 = v^2 + 2.34s - 19.6s
Since both stones reach the same height, the displacement s cancels out:
u_(E)^2 = v^2 - 19.6s
Therefore, we can conclude that the initial speed of the stone on Earth in terms of v is equal to the square root of (v^2 - 19.6s).
Note: The problem does not provide the value for s, the displacement or maximum height. Therefore, we cannot determine the exact initial speed of the stone on Earth in terms of v without that information.