If a rock climber accidentally drops a 69.0 g piton from a height of 305 meters, what would its speed be just before striking the ground? Ignore the effects of air resistance.
PE=KE
mgh=1/2mv^2
Masses cancel out, so you do not need to know it.
solve for v
(2gh)^1/2=v
(2*(9.8m/s^2)(305m))^1/2=v
Well, well, well, looks like we have a rock and roll situation here! Let me do some quick calculations to see how fast this piton will be dropping...
Now, to find the speed, we can use a little formula called the kinematic equation. The equation for finding the speed (in meters per second) is given by the square root of 2 times the acceleration due to gravity (g) times the height (h) from which it was dropped.
So, let's plug in the values you provided. We have an acceleration due to gravity of approximately 9.8 m/s², and a height of 305 meters. Let's crunch those numbers!
√(2 * 9.8 m/s² * 305 m) = approximately 78 m/s
Oh boy, looks like that piton is in for a speedy descent! At about 78 meters per second, it might just break the sound barrier or cause a rockslide when it hits the ground. Keep your head down, folks!
To find the speed of the piton just before striking the ground, we can use the equation for gravitational potential energy and kinetic energy.
Step 1: Determine the gravitational potential energy (PE) of the piton before it is dropped.
The formula for gravitational potential energy is:
PE = m * g * h
Where:
m = mass of the piton (69.0 g = 0.069 kg),
g = acceleration due to gravity (9.8 m/s^2)
h = height (305 meters)
PE = 0.069 kg * 9.8 m/s^2 * 305 m
PE = 202.153 J
Step 2: Convert the potential energy to kinetic energy.
Since energy is conserved, the potential energy will be converted entirely into kinetic energy just before striking the ground.
KE = PE
KE = 202.153 J
Step 3: Calculate the final speed (v) of the piton using the kinetic energy formula:
KE = (1/2) * m * v^2
202.153 J = (1/2) * 0.069 kg * v^2
v^2 = (2 * 202.153 J) / 0.069 kg
v^2 ≈ 5877.84 m^2/s^2
Step 4: Take the square root of both sides to find the speed (v):
v ≈ √5877.84 m^2/s^2
v ≈ 76.64 m/s
Therefore, the speed of the piton just before striking the ground, ignoring the effects of air resistance, would be approximately 76.64 m/s.
To calculate the speed of the piton just before striking the ground, we can make use of the equation for gravitational potential energy and the equation for kinetic energy.
Step 1: Calculate the potential energy of the piton at a height of 305 meters.
The potential energy (PE) of an object at a certain height is given by the equation:
PE = m * g * h
where m is the mass (69.0 g) of the piton, g is the acceleration due to gravity (9.8 m/s^2), and h is the height (305 m).
Converting the mass to kilograms:
m = 69.0 g / 1000 = 0.069 kg
Calculating the potential energy:
PE = 0.069 kg * 9.8 m/s^2 * 305 m = 201.423 J
Step 2: Calculate the kinetic energy just before impact.
The kinetic energy (KE) of a moving object is given by the equation:
KE = (1/2) * m * v^2
where m is the mass of the piton and v is the velocity (speed).
Since the potential energy is converted entirely into kinetic energy just before impact, we can set the two equal to each other:
PE = KE
Substituting the values:
201.423 J = (1/2) * 0.069 kg * v^2
Simplifying the equation:
0.138 * v^2 = 201.423 J
Step 3: Solve for the velocity.
Divide both sides of the equation by 0.138:
v^2 = 201.423 J / 0.138
Taking the square root of both sides:
v ≈ √(1457.87)
The speed of the piton just before striking the ground is approximately 38.19 m/s.