You are standing at the top of a cliff that has
a stairstep configuration. There is a vertical
drop of 6 m at your feet, then a horizontal
shelf of 10 m , then another drop of 4 m to the
bottom of the canyon, which has a horizontal
floor. You kick a 0.49 kg rock, giving it an
initial horizontal velocity that barely clears
the shelf below.What initial horizontal velocity v will be
required to barely clear the edge of the shelf
below you? The acceleration of gravity is
9.8 m/s
2
. Consider air friction to be negligible.
Answer in units of m/s.
To determine the initial horizontal velocity required for the rock to barely clear the edge of the shelf below, we can use the principle of conservation of energy.
The potential energy at the top of the cliff is converted to kinetic energy as the rock falls. The energy at the top can be calculated using the formula:
Potential Energy = m * g * h,
where m is the mass of the rock (0.49 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the vertical drop (6 m).
Therefore, the potential energy at the top of the cliff is:
Potential Energy = 0.49 kg * 9.8 m/s^2 * 6 m = 28.728 J.
The kinetic energy of the rock just before it reaches the edge of the shelf is equal to the potential energy at the top. So, the kinetic energy can be calculated using the formula:
Kinetic Energy = (1/2) * m * v^2,
where m is the mass of the rock (0.49 kg) and v is the initial horizontal velocity.
By equating the potential and kinetic energy, we get:
28.728 J = (1/2) * 0.49 kg * v^2.
Simplifying the equation, we have:
57.456 J = 0.49 kg * v^2.
Dividing both sides by 0.49 kg, we find:
117 m^2/s^2 = v^2.
Taking the square root of both sides, we get:
v = sqrt(117) m/s.
Therefore, the initial horizontal velocity required to barely clear the edge of the shelf below is approximately 10.82 m/s.
To find the initial horizontal velocity required to barely clear the edge of the shelf below, we can use the principle of conservation of energy.
We can break the motion of the rock into two parts: the vertical motion and the horizontal motion.
1. Vertical Motion:
The rock experiences a vertical displacement of 6 m initially, then another 4 m. The total vertical displacement is 6 m + 4 m = 10 m. The rock will reach its highest point when all of its initial potential energy is converted to kinetic energy. At this point, the rock has no potential energy, only kinetic energy.
The potential energy at the top of the cliff is given by: mgh, where m is the mass of the rock (0.49 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the vertical displacement (10 m).
Potential energy at the top = mgh = (0.49 kg)(9.8 m/s^2)(10 m) = 48.02 J
At the highest point, this potential energy is converted entirely into kinetic energy:
Kinetic energy at the highest point = 48.02 J
2. Horizontal Motion:
To find the initial horizontal velocity required, we need to calculate the horizontal kinetic energy of the rock at the highest point.
The horizontal kinetic energy is given by: 1/2 * m * v^2, where m is the mass of the rock (0.49 kg) and v is the initial horizontal velocity we're trying to find.
Equating the horizontal kinetic energy to the potential energy at the highest point, we get:
1/2 * m * v^2 = 48.02 J
Simplifying the equation:
1/2 * 0.49 kg * v^2 = 48.02 J
(0.245 kg) * v^2 = 48.02 J
v^2 = 48.02 J / (0.245 kg)
v^2 ≈ 196.16 m^2/s^2
Taking the square root of both sides to solve for the initial horizontal velocity:
v ≈ √(196.16 m^2/s^2)
v ≈ 14.00 m/s
So, the initial horizontal velocity (v) required to barely clear the edge of the shelf below is approximately 14.00 m/s.