A roller coaster cart is pulled up to the top of the hill 45 m high before it plunges down to its lowest point, 5 m above the ground. What is the cart’s speed at this point? What is its potential energy if the mass of the object is 5 kg?
(1/2) m v^2 = m g (45-5)
v^2 = 2 (9.81)(40)
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Second question has no real answer unless we assume that the potential energy is zero at the ground level.
ONLY changes in potential have meaning,and you must always define a base
If we ASSUME U = 0 at
ground
then at 5 meters up
U = potential energy = m g h
= 5 9.81 * 5 Joules
Well, let's calculate that!
First, let's find the potential energy at the highest point.
Potential energy (PE) = mass (m) × gravity (g) × height (h)
PE = 5 kg × 9.8 m/s² × 45 m
PE = 2205 J
Now, let's use the principle of conservation of energy to find the speed at the lowest point.
The potential energy at the highest point is converted into kinetic energy at the lowest point.
Potential energy (PE) = Kinetic energy (KE)
So, 2205 J = 1/2 × mass (m) × velocity (v)²
Since the mass is given as 5 kg, we have:
2205 J = 1/2 × 5 kg × v²
Now, let's solve for v:
v² = 2 × (2205 J / 5 kg)
v² = 2 × 441 J/kg
v² = 882 J/kg
Taking the square root of both sides, we get:
v = √882 J/kg
And since we're dealing with a roller coaster, we know we want the speed, not the velocity. So let's keep only the positive value:
v = √882 J/kg ≈ 29.67 m/s
So, the speed of the cart at the lowest point is approximately 29.67 m/s.
Now, if you've got any more questions or need a joke to lighten the mood, just let me know!
To find the speed of the roller coaster cart at its lowest point, we can use the principle of conservation of energy. At the top of the hill, the cart has potential energy that will be converted into kinetic energy as it moves downhill.
1. The potential energy of an object is given by the formula: PE = m * g * h, where PE is the potential energy, m is the mass, g is the acceleration due to gravity (approximately 9.8 m/s²), and h is the height.
Given that the mass of the object is 5 kg and the height above the ground is 45 m, we can calculate the potential energy at the top:
PE = 5 kg * 9.8 m/s² * 45 m
PE = 2205 J
2. At the lowest point, all the potential energy is converted into kinetic energy. The formula for kinetic energy is KE = (1/2) * m * v², where KE is the kinetic energy and v is the velocity or speed.
Since we know that the potential energy at the top is converted into kinetic energy at the bottom, we can equate the two:
PE = KE
2205 J = (1/2) * 5 kg * v²
3. Rearranging the equation to solve for v, we get:
v = sqrt((2 * PE) / m)
v = sqrt((2 * 2205 J) / 5 kg)
v = sqrt(4410 J / 5 kg)
v = sqrt(882 m²/s²)
v ≈ 29.7 m/s
Therefore, the speed of the roller coaster cart at its lowest point is approximately 29.7 m/s.
To find the potential energy of the object, we can use the formula:
PE = m * g * h, where PE is the potential energy, m is the mass, g is the acceleration due to gravity, and h is the height.
Given that the mass of the object is 5 kg and the height above the ground is 45 m, we can calculate the potential energy as follows:
PE = 5 kg * 9.8 m/s² * 45 m
PE = 2205 J
Therefore, the potential energy of the object is 2205 Joules.
To find the speed of the roller coaster cart at its lowest point, we can use the principle of conservation of energy. At the highest point (45 m), the cart has potential energy which will be converted into kinetic energy at the lowest point (5 m). We can equate these two energies to find the speed.
1. Calculate potential energy at the highest point:
Potential energy (PE) = mass (m) x gravity (g) x height (h)
PE = 5 kg x 9.8 m/s^2 x 45 m
2. Calculate potential energy at the lowest point:
PE = 5 kg x 9.8 m/s^2 x 5 m
Since potential energy is being converted into kinetic energy, we can equate the two:
PE at highest point = PE at lowest point
5 kg x 9.8 m/s^2 x 45 m = 5 kg x 9.8 m/s^2 x 5 m
Simplifying the equation, we can cancel out the mass and gravity:
45 m = 5 m
Since the heights on both sides are equal, it means the cart has lost all its potential energy and gained kinetic energy. Therefore, the speed of the cart at this point is:
Speed = sqrt(2 x gravity x height)
Speed = sqrt(2 x 9.8 m/s^2 x 5 m)
Calculating this, we get:
Speed ≈ 9.9 m/s
Now let's calculate the potential energy of the cart when it is 5 m above the ground:
Potential energy (PE) = mass (m) x gravity (g) x height (h)
PE = 5 kg x 9.8 m/s^2 x 5 m
Calculating this, we get:
PE ≈ 245 J (joules)
So, the speed of the cart at its lowest point is approximately 9.9 m/s, and its potential energy at this point is approximately 245 J, given the mass of the object is 5 kg.