A rock of mass 5kg is pushed over the edge of a cliff which is 200m high
1.ignore air resistance and calculate the speed of the rock at the bottom of the cliff. In reality air resistance cannot be ignored. The rock eventually reaches terminal velocity of 30m/s
2.calculate the kinetic energy of the rock at the bottom of the cliff
3.calculate work done by air resistance
4.calculate the average force exerted by air on the rock
1. We can use the following equation of motion to calculate the speed of the rock at the bottom of the cliff, ignoring air resistance:
v^2 = u^2 + 2as
where v is the final velocity, u is the initial velocity (0 m/s in this case as the rock is initially at rest), a is acceleration due to gravity (approximately 9.81 m/s^2), and s is the displacement (200 m).
v^2 = 0 + 2 * 9.81 * 200
v^2 = 3924
v = sqrt(3924) = 62.6 m/s (approximately)
2. Now we can calculate the kinetic energy of the rock at the bottom of the cliff. Kinetic energy is given by the equation:
KE = 1/2 * m * v^2
where m is the mass of the rock (5 kg) and v is its final velocity (62.6 m/s, approximately).
KE = 1/2 * 5 * (62.6)^2 = 0.5 * 5 * 3918.76 = 9789.4 J (approximately)
3. Since we know that the rock eventually reaches a terminal velocity of 30 m/s, we can calculate the kinetic energy at this point using the same equation.
KE_terminal = 1/2 * m * v_terminal^2
KE_terminal = 1/2 * 5 * (30)^2 = 0.5 * 5 * 900 = 2250 J
Now we can find the work done by air resistance by finding the difference between the kinetic energy without air resistance and the kinetic energy at terminal velocity.
Work_air = KE - KE_terminal = 9789.4 - 2250 = 7539.4 J (approximately)
4. To calculate the average force exerted by air on the rock, we can use the work-energy principle:
Work = Force x distance
In this case, the distance is the height of the cliff, which is 200 m. We can rearrange the equation to solve for the force:
Force = Work / distance = 7539.4 / 200 = 37.7 N (approximately)
So the average force exerted by air on the rock is 37.7 N.
1. Well, let's assume this rock is an overachiever and can defy the laws of physics by completely ignoring the pesky air resistance. In that case, we can use good old kinematic equations to calculate its speed at the bottom of the cliff. We know that the initial velocity is zero (since it's pushed) and the acceleration due to gravity is about 9.8 m/s². Using the equation s = ut + (1/2)at², where s is the distance, u is the initial velocity, a is the acceleration, and t is the time, we can plug in the values. The distance fallen is 200m, initial velocity u is 0, acceleration a is 9.8 m/s², and we want to find the time t it takes for it to reach the bottom. Rearranging the equation, we get t = √(2s/a). Plugging in the values, we get t = √(2(200)/9.8) ≈ 6.43 seconds. Given all this information, the speed of the rock at the bottom of the cliff is the final velocity. So, using the equation v = u + at, where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time, let's calculate it. v = 0 + (9.8)(6.43) ≈ 63.19 m/s. Voila!
2. Well, now that we know the speed of the rock at the bottom of the cliff (approximately 63.19 m/s), we can calculate its kinetic energy. The formula for kinetic energy is KE = (1/2)mv², where KE is the kinetic energy, m is the mass of the rock, and v is its velocity. Plugging in the values, we get KE = (1/2)(5)(63.19)² ≈ 7,968.97 Joules. So, the kinetic energy of the rock at the bottom of the cliff is approximately 7,968.97 Joules. That's a lot of energy to rock and roll!
3. Ah, now it's time to deal with the rebellious air resistance! In order to calculate the work done by air resistance, we need to know the displacement of the rock. Since it falls straight down, the displacement is the height of the cliff, which is 200m. The work done is given by the equation W = Fd, where W is the work done, F is the force, and d is the displacement. But since the force is opposite to the displacement, the work done by air resistance is negative. Oh, air resistance, always bringing negativity into our lives! Since we know the rock reaches terminal velocity of 30m/s, we can use the equation KE = (1/2)mv² to calculate the force of air resistance. Rearranging the equation, we get F = (2KE)/(v²). Plugging in the values, we get F = (2(7,968.97))/(30²) ≈ 17,493.93 Newtons. And since the work done is negative, the work done by air resistance is W = (-F)d = (-17,493.93)(200) ≈ -3,498,786.4 Joules. Negative work, such a party pooper!
4. Now let's calculate the average force exerted by air on the rock. We know the work done is negative (-3,498,786.4 Joules) and the displacement is 200m. The equation for average force is F = W/d, where F is the average force, W is the work done, and d is the displacement. Plugging in the values, we get F = (-3,498,786.4)/(200) ≈ -17,493.93 Newtons. Negative force? Guess air resistance isn't a huge fan of rockstars falling from cliffs!
1. To calculate the speed of the rock at the bottom of the cliff without considering air resistance, we can use the principle of conservation of energy.
The potential energy of the rock at the top of the cliff is given by:
Potential energy = mass * acceleration due to gravity * height
Potential energy = 5 kg * 9.8 m/s^2 * 200 m
Next, this potential energy is converted into kinetic energy at the bottom of the cliff:
Kinetic energy = (1/2) * mass * velocity^2
Equating potential energy and kinetic energy,
5 kg * 9.8 m/s^2 * 200 m = (1/2) * 5 kg * v^2
Simplifying the equation:
9800 N * 200 m = (1/2) * 5 kg * v^2
1960000 Nm = 2.5 kg * v^2
Solving for velocity (v):
v^2 = (1960000 Nm) / (2.5 kg)
v = sqrt((1960000 Nm) / (2.5 kg))
v ≈ 88.39 m/s
Therefore, the speed of the rock at the bottom of the cliff, neglecting air resistance, is approximately 88.39 m/s.
2. To calculate the kinetic energy of the rock at the bottom of the cliff, we use the formula:
Kinetic energy = (1/2) * mass * velocity^2
Kinetic energy = (1/2) * 5 kg * (88.39 m/s)^2
Simplifying the equation:
Kinetic energy = (1/2) * 5 kg * 7821.5121 m^2/s^2
Kinetic energy ≈ 19553 Nm
Therefore, the kinetic energy of the rock at the bottom of the cliff is approximately 19553 Nm.
3. To calculate the work done by air resistance, we need to know the distance traveled by the rock and the force exerted by air resistance.
However, since the rock reaches terminal velocity, the force exerted by air resistance is equal and opposite to the gravitational force acting on the rock (weight). Therefore, the net force acting on the rock is zero, and no work is done by air resistance.
Work done by air resistance = 0 Joules
4. Since no work is done by air resistance, the average force exerted by the air on the rock is also zero.
Average force exerted by air = 0 Newtons
1. To calculate the speed of the rock at the bottom of the cliff without air resistance, you can use the principle of conservation of energy. The potential energy of the rock at the top of the cliff is converted into kinetic energy at the bottom.
First, calculate the potential energy of the rock at the top of the cliff:
Potential Energy = mass * acceleration due to gravity * height
Potential Energy = 5kg * 9.8m/s^2 * 200m
Next, equate the potential energy to the kinetic energy at the bottom:
Potential Energy = Kinetic Energy
5kg * 9.8m/s^2 * 200m = 1/2 * mass * velocity^2
Solve for velocity:
velocity = sqrt((2 * Potential Energy) / mass)
velocity = sqrt((2 * (5kg * 9.8m/s^2 * 200m)) / 5kg)
Calculate the velocity to find the answer.
However, since air resistance cannot be ignored in reality, the rock will reach a terminal velocity of 30m/s. This means that the rock will not continue to accelerate and its speed will remain constant at 30m/s.
2. To calculate the kinetic energy of the rock at the bottom of the cliff, you can use the equation:
Kinetic Energy = 1/2 * mass * velocity^2
Plug in the given values for the mass and velocity, and calculate the kinetic energy.
3. The work done by air resistance can be calculated by multiplying the force of air resistance by the distance traveled. Since the force of air resistance depends on the velocity of the object, you will need to break down the problem into smaller intervals to account for the change in velocity as the rock accelerates due to gravity and then reaches its terminal velocity.
4. The average force exerted by air on the rock can be calculated by dividing the work done by air resistance by the distance traveled.