A roller coaster of mass 1000.0 kg passes point A with a speed of 1.80 m/s?
point a= 30m
point b= 0m
point c= 25m
point d= 12m
1. What is the total mechanical energy of the roller coaster at point A?
2. What is the speed of the roller coaster at point B?
3. What is the potential energy and the kinetic energy of the roller coaster at point C?
4. Derive an expression for the speed of the cart using the variables E, m, g, and h.
5. Determine the speed of the roller coaster at D.
reference potential energy to ground
U = m g h
Ke = (1/2) m v^2
1
E = U + Ke = 1000*9.8*30 + 500 (1.8)^2
2
E = 0 + Ke
so (1/2) m (Vat B)^2 = E from 1 above
3
same old E from part 1
E = 1000*9.8*25 + Ke at C
4
(1/2) m v^2 = E - m g h
5
(1/2) m v^2 = E - m g h
is damon answer right??
1. To find the total mechanical energy of the roller coaster at point A, we need to consider both its potential energy and kinetic energy. The equation for total mechanical energy (E) is given by:
E = potential energy (PE) + kinetic energy (KE)
Since the roller coaster is at a height above the ground at point A, its potential energy can be calculated using the equation:
PE = m * g * h
where m is the mass of the roller coaster (1000.0 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the height above the ground at point A (30 m).
Calculating the potential energy:
PE = 1000.0 kg * 9.8 m/s^2 * 30 m = 294000 J
The kinetic energy of the roller coaster can be calculated using the equation:
KE = (1/2) * m * v^2
where v is the speed of the roller coaster at point A (1.80 m/s).
Calculating the kinetic energy:
KE = (1/2) * 1000.0 kg * (1.80 m/s)^2 = 1620 J
Therefore, the total mechanical energy of the roller coaster at point A is:
E = 294000 J + 1620 J = 295620 J
2. To find the speed of the roller coaster at point B, we can use the principle of conservation of mechanical energy. This principle states that total mechanical energy is conserved in a closed system, meaning the sum of potential and kinetic energy remains constant throughout the motion.
Since we have already calculated the total mechanical energy at point A as 295620 J, we can use it to find the kinetic energy at point B:
E = KE at B
(1/2) * m * v^2 = 295620 J
Solving for v:
v^2 = (2 * 295620 J) / 1000.0 kg
v^2 = 591.24 m^2/s^2
v = sqrt(591.24 m^2/s^2) = 24.32 m/s
Therefore, the speed of the roller coaster at point B is 24.32 m/s.
3. At point C, the roller coaster is at a height of 25 m from the ground. To determine its potential energy and kinetic energy at this point:
Potential Energy at C (PE) = m * g * h
= 1000.0 kg * 9.8 m/s^2 * 25 m
= 245000 J
Kinetic Energy at C (KE) = (1/2) * m * v^2
= (1/2) * 1000.0 kg * (1.80 m/s)^2
= 1620 J
Therefore, the potential energy of the roller coaster at point C is 245000 J and the kinetic energy is 1620 J.
4. Deriving an expression for the speed (v) of the roller coaster using the variables E (total mechanical energy), m (mass of the roller coaster), g (acceleration due to gravity), and h (height above the ground).
At any point in the roller coaster's motion, the total mechanical energy (E) is given by:
E = m * g * h + (1/2) * m * v^2
Rearranging the equation, we find:
(1/2) * m * v^2 = E - m * g * h
Using the definition of potential energy (PE = m * g * h), we can rewrite the equation as:
(1/2) * m * v^2 = E - PE
Simplifying further:
v^2 = (2 * (E - PE)) / m
Taking the square root of both sides:
v = sqrt((2 * (E - PE)) / m)
Therefore, the expression for the speed of the roller coaster (v) in terms of the variables E, m, g, and h is:
v = sqrt((2 * (E - m * g * h)) / m)
5. To determine the speed of the roller coaster at point D, we can use the equation derived in the previous step:
v = sqrt((2 * (E - m * g * h)) / m)
Given the total mechanical energy at point A as 295620 J, and the potential energy and kinetic energy at point D can be calculated as follows:
Potential Energy at D (PE) = m * g * h
= 1000.0 kg * 9.8 m/s^2 * 12 m
= 117600 J
Using these values, we can now calculate the speed of the roller coaster at point D:
v = sqrt((2 * (295620 J - 1000.0 kg * 9.8 m/s^2 * 12 m)) / 1000.0 kg)
Simplifying and calculating:
v = sqrt((591240 J - 117600 J) / 1000.0 kg)
= sqrt(473640 J / 1000.0 kg)
= sqrt(473.64 m^2/s^2)
= 21.77 m/s
Therefore, the speed of the roller coaster at point D is 21.77 m/s.
1. To calculate the total mechanical energy at point A, we need to consider both the potential energy and the kinetic energy.
The potential energy at point A can be calculated using the formula:
Potential Energy = mass * gravitational acceleration * height
Potential Energy at point A = 1000.0 kg * 9.8 m/s^2 * 30 m
The kinetic energy at point A can be calculated using the formula:
Kinetic Energy = (1/2) * mass * velocity^2
Kinetic Energy at point A = 0.5 * 1000.0 kg * (1.80 m/s)^2
Total Mechanical Energy at point A = Potential Energy at point A + Kinetic Energy at point A
2. To find the speed of the roller coaster at point B, we need to apply the principle of conservation of energy. According to this principle, the total mechanical energy remains constant throughout the roller coaster's motion.
Since we know the total mechanical energy at point A and point B is at a lower height than point A (meaning it has less potential energy), we can equate the two expressions:
Potential Energy at point B + Kinetic Energy at point B = Total Mechanical Energy at point A
As we are given the height at point B is 0m, the potential energy at point B becomes zero. Therefore, we can rewrite the equation as:
Kinetic Energy at point B = Total Mechanical Energy at point A
To find the speed at point B, we can use the formula for kinetic energy and solve for velocity:
Kinetic Energy = (1/2) * mass * velocity^2
0.5 * 1000.0 kg * (velocity at B)^2 = Total Mechanical Energy at point A
Solve for velocity at B.
3. To determine the potential energy and kinetic energy at point C, we repeat the process as we did for point A.
The potential energy at point C can be calculated using the formula:
Potential Energy = mass * gravitational acceleration * height
Potential Energy at point C = 1000.0 kg * 9.8 m/s^2 * 25 m
The kinetic energy at point C can be calculated using the formula:
Kinetic Energy = (1/2) * mass * velocity^2
We need the velocity at point C to calculate the kinetic energy.
4. To derive an expression for the speed of the cart using the variables E, m, g, and h, we can use the principle of conservation of energy.
Starting with the total mechanical energy equation:
Total Mechanical Energy = Potential Energy + Kinetic Energy
Potential Energy = m * g * h (mass * gravitational acceleration * height)
Kinetic Energy = (1/2) * m * v^2 (1/2 * mass * velocity^2)
Substituting these equations into the total mechanical energy equation, we get:
E = m * g * h + (1/2) * m * v^2
Solve this equation for v (velocity) to get the expression for the speed of the cart.
5. To determine the speed of the roller coaster at point D, we need to use the principle of conservation of energy again.
As we know that mechanical energy is conserved, the total mechanical energy at point C will be equal to the total mechanical energy at point D.
We already know the potential energy at point C and can calculate the kinetic energy at point C using the given information:
Kinetic Energy at point C = (1/2) * 1000.0 kg * (velocity at C)^2
To find the speed at point D, we equate the total mechanical energy at point C with the total mechanical energy at point D:
Potential Energy at point C + Kinetic Energy at point C = Potential Energy at point D + Kinetic Energy at point D
Given that the height at point D is 12m, we can calculate the potential energy at point D using the formula:
Potential Energy at point D = 1000.0 kg * 9.8 m/s^2 * 12 m
Using the above equation, solve for the velocity at point D.