Sheila and her toboggan have a total mass of 45 kg. At one point during her toboggan ride down a frictionless hill, she is 10.0m higher than the bottom of the hill and moving at a speed of 8.0m/s. (5 marks)
a) How much gravitational potential energy does she have?
b) How much kinetic energy does she have?
c) How much energy does she have in total?
d) How fast will Sheila be moving when she reaches the bottom of the hill? e) How high is the hill?
STEPS PLEASE!
a) m g h ... 45 kg * 9.8 m/s^2 * 10.0 m = ? Joules
b) 1/2 m v^2 = 1/2 * 45 kg * (8.0 m/s)^2 = ? Joules
c) add the two energies
d) her total energy will be all kinetic
... v = √[2 * (total energy) / m]
e) m g h = total energy
... h = (total energy) / (m * g)
a) Sheila's gravitational potential energy can be calculated using the formula: Potential Energy = mass * gravity * height.
Given:
Mass (m) = 45 kg
Gravity (g) = 9.8 m/s^2
Height (h) = 10.0 m
Potential Energy = 45 kg * 9.8 m/s^2 * 10.0 m = 4410 J
Therefore, Sheila has 4410 Joules of gravitational potential energy.
b) Sheila's kinetic energy can be calculated using the formula: Kinetic Energy = 0.5 * mass * velocity^2.
Given:
Mass (m) = 45 kg
Velocity (v) = 8.0 m/s
Kinetic Energy = 0.5 * 45 kg * (8.0 m/s)^2 = 1440 J
Therefore, Sheila has 1440 Joules of kinetic energy.
c) The total energy of the system is equal to the sum of potential energy and kinetic energy.
Total Energy = Potential Energy + Kinetic Energy
Total Energy = 4410 J + 1440 J = 5850 J
Therefore, Sheila has a total energy of 5850 Joules.
d) To find Sheila's speed at the bottom of the hill, we can use the principle of conservation of energy, which states that the total mechanical energy (sum of potential and kinetic energy) is conserved throughout the ride.
Since there is no friction, the total energy at the top of the hill is equal to the total energy at the bottom of the hill.
Initial total energy (at the top) = Final total energy (at the bottom)
Potential Energy (at the top) + Kinetic Energy (at the top) = Potential Energy (at the bottom) + Kinetic Energy (at the bottom)
Potential Energy (at the top) = 4410 J (from part a)
Kinetic Energy (at the top) = 1440 J (from part b)
Potential Energy (at the bottom) = 0 J (since the bottom is the reference point)
Let's denote the final kinetic energy as Kf, and the final velocity as vf.
4410 J + 1440 J = 0 J + Kf
5850 J = Kf
Therefore, Sheila will have a kinetic energy of 5850 Joules at the bottom of the hill.
Finally, using the formula for kinetic energy:
Kf = 0.5 * mass * velocity^2
5850 J = 0.5 * 45 kg * vf^2
vf^2 = 5850 J * 2 / 45 kg
vf^2 = 130 J/kg
vf = sqrt(130 J/kg) ≈ 11.4 m/s
Therefore, Sheila will be moving at approximately 11.4 m/s when she reaches the bottom of the hill.
e) Since the final kinetic energy is determined solely by the height difference, we can determine the height of the hill using the equation:
Final Kinetic Energy (at the bottom) = mass * gravity * height
From part b, the final kinetic energy is 1440 J.
1440 J = 45 kg * 9.8 m/s^2 * height
height = 1440 J / (45 kg * 9.8 m/s^2)
height ≈ 3.14 m
Therefore, the height of the hill is approximately 3.14 meters.
Sure! Let's go step-by-step to solve this problem:
a) To find the gravitational potential energy (PE), we can use the formula PE = mgh, where m is the mass, g is the acceleration due to gravity (9.8 m/s^2), and h is the height.
Given:
- Mass (m) = 45 kg
- Height (h) = 10.0 m
- Acceleration due to gravity (g) = 9.8 m/s^2
Substituting these values into the formula, we get:
PE = (45 kg) * (9.8 m/s^2) * (10.0 m)
= 4410 J
Therefore, Sheila has 4410 J of gravitational potential energy.
b) To find the kinetic energy (KE), we can use the formula KE = 0.5 * m * v^2, where m is the mass and v is the velocity.
Given:
- Mass (m) = 45 kg
- Velocity (v) = 8.0 m/s
Substituting these values into the formula, we get:
KE = 0.5 * (45 kg) * (8.0 m/s)^2
= 1440 J
Therefore, Sheila has 1440 J of kinetic energy.
c) The total energy of the system is the sum of the potential energy (PE) and kinetic energy (KE). Therefore, the total energy (TE) is given by:
TE = PE + KE
= 4410 J + 1440 J
= 5850 J
So, Sheila has a total energy of 5850 J.
d) To find out how fast Sheila will be moving when she reaches the bottom of the hill, we can use the principle of conservation of energy. The total mechanical energy (TM) at the top equals the total mechanical energy at the bottom.
At the top of the hill, Sheila has potential energy (PE) and kinetic energy (KE). At the bottom of the hill, Sheila only has kinetic energy (KE) since the potential energy has been converted to kinetic energy.
Therefore, at the bottom of the hill, the total mechanical energy is equal to the kinetic energy:
TM = KE = 1440 J
To find Sheila's velocity when she reaches the bottom of the hill, we can use the formula KE = 0.5 * m * v^2.
Given:
- Mass (m) = 45 kg
- Total mechanical energy (TM) = 1440 J
Rearranging the formula, we get:
TM = 0.5 * m * v^2
1440 J = 0.5 * (45 kg) * v^2
v^2 = (1440 J) * 2 / (45 kg)
v^2 = 64
v = √64
v = 8 m/s
Therefore, Sheila will be moving at a speed of 8.0 m/s when she reaches the bottom of the hill.
e) Given that Sheila is 10.0 m higher than the bottom of the hill and the bottom of the hill is at zero height, the height of the hill can be calculated as the difference between Sheila's height and the bottom of the hill.
Height of the hill = Sheila's height - Bottom of the hill
= 10.0 m - 0 m
= 10.0 m
Therefore, the height of the hill is 10.0 m.
Sure! Let's go through the steps to find the answers to each question:
a) To find the gravitational potential energy, we need to use the formula:
Gravitational Potential Energy = mass * gravity * height
Here, the mass of Sheila and the toboggan is given as 45 kg. The acceleration due to gravity can be taken as 9.8 m/s^2, and the height is given as 10.0 m. Using the formula, we have:
Gravitational Potential Energy = 45 kg * 9.8 m/s^2 * 10.0 m
= 4410 J
So, Sheila has 4410 J of gravitational potential energy.
b) To find the kinetic energy, we need to use the formula:
Kinetic Energy = 0.5 * mass * velocity^2
Here, the mass is still 45 kg and the velocity is given as 8.0 m/s. Plugging in these values, we have:
Kinetic Energy = 0.5 * 45 kg * (8.0 m/s)^2
= 1440 J
So, Sheila has 1440 J of kinetic energy.
c) To find the total energy, we simply add the gravitational potential energy and the kinetic energy together:
Total Energy = Gravitational Potential Energy + Kinetic Energy
= 4410 J + 1440 J
= 5850 J
So, Sheila has a total energy of 5850 J.
d) To find Sheila's speed at the bottom of the hill, we can use the principle of conservation of energy. Since the hill is frictionless, the total energy remains constant throughout the ride. So we can equate the initial potential energy (at the top of the hill) to the final kinetic energy (at the bottom of the hill):
Gravitational Potential Energy = Kinetic Energy
We can re-arrange the formula for kinetic energy to solve for velocity:
Kinetic Energy = 0.5 * mass * velocity^2
Velocity = √(2 * Kinetic Energy / mass)
Using the values we already know, the mass is 45 kg and the Kinetic Energy is 1440 J, we can calculate:
Velocity = √(2 * 1440 J / 45 kg)
≈ √(64 m^2/s^2)
≈ 8.0 m/s
So, Sheila will be moving at 8.0 m/s when she reaches the bottom of the hill.
e) To find the height of the hill, we already have the given value of 10.0 m.
Therefore, the height of the hill is 10.0 m.
I hope this explanation helps you understand how to find the answers step-by-step! Let me know if there's anything else I can assist you with.