If a rock climber accidentally drops a 56-gram piton from a height of 375 meters, what would its speed be before striking the ground? Ignore the effects of air resistance.
I think you can get this from
distance = (1/2)gt^2 and solve for t, then
distance = (1/2)vt
Solve for v.
45
To determine the speed of the piton before striking the ground, we can use the equation of motion:
\[ v^2 = u^2 + 2as \]
where:
- \( v \) represents the final velocity (which is what we want to find),
- \( u \) represents the initial velocity (which is 0 because the piton was dropped without an initial velocity),
- \( a \) represents acceleration due to gravity (approximately \( 9.8 \, \text{m/s}^2 \)),
- \( s \) represents the distance traveled by the piton (which is the height from which it was dropped, 375 meters).
Substituting the known values into the equation, we have:
\[ v^2 = 0^2 + 2 \cdot 9.8 \cdot 375 \]
Simplifying further:
\[ v^2 = 0 + 7350 \]
\[ v = \sqrt{7350} \]
Calculating:
\[ v \approx 85.74 \, \text{m/s} \]
Therefore, the speed of the piton before striking the ground would be approximately 85.74 m/s.
To determine the speed of the piton before it strikes the ground, we can use the principle of conservation of energy. The potential energy possessed by the piton when it is at a certain height is converted to kinetic energy when it reaches the ground.
The potential energy of an object at a certain height is given by the formula:
Potential Energy = mass x acceleration due to gravity x height
In this case, the mass of the piton is 56 grams, which is equivalent to 0.056 kilograms. The acceleration due to gravity is approximately 9.8 meters per second squared. The height is 375 meters.
Potential Energy = 0.056 kg x 9.8 m/s^2 x 375 m
Potential Energy = 206.55 Joules
Since there is no air resistance present, the potential energy is completely converted to kinetic energy as the piton falls.
The equation for kinetic energy is:
Kinetic Energy = (1/2) x mass x velocity^2
We can rearrange this equation to solve for velocity:
Velocity = √[(2 x Kinetic Energy) / mass]
Substituting the potential energy calculated earlier for the kinetic energy, and the mass of the piton, we get:
Velocity = √[(2 x 206.55 J) / 0.056 kg]
Velocity = √[7357.14 m^2/s^2]
Velocity = 85.78 m/s
Therefore, the speed of the piton before striking the ground is approximately 85.78 meters per second.