A 4kg block is sitting on the floor.
(a) How much potential energy does it have? (b) How much kinetic energy does it have?
(c) The block is raised to 2m high. How much potential energy does it have? (d) The block is raised to 4m high. How much potential energy does it have?
(e) The block is raised to 45m high. How much potential energy does it have? (f) The block is dropped to the ground. How fast is it traveling when it hits the ground? (remember chapter 2?)
(g) How much kinetic energy does it have when it hits the ground?
(a)
recall that potential energy is stored energy, and is given by the formula:
PE = mgh (units in Joules)
where
m = mass (in kg)
g = acceleration due to gravity = 9.8 m/s^2
h = height (in m)
if the reference position is at the floor (that is, h = 0), the PE is equal to
PE = 4*9.8*0
PE = 0
(b)
recall that kinetic energy is energy in motion and is given by the formula:
KE = (1/2)*m*v^2
where
v = velocity (in m/s)
since it's not moving (v = 0),
KE = 0
(c)
at 2 m high,
PE = 4*9.8*2
PE = 78.4 J
(d)
at 4 m high,
PE = 4*9.8*4
PE = 158.8 J
(e)
at 45 m high,
PE = 4*9.8*45
PE = 1764 J
(f)
recall that the motion of the block is uniformly accelerated motion, and we can therefore use the formula:
(v,f)^2 - (v,o)^2 = 2gh
where
v,f = final velocity (in m/s)
v,o = initial velocity (in m/s)
since it is dropped from rest, v,o = 0:
v,f^2 = 2gh
v,f = sqrt(2gh)
v,f = sqrt(2*9.8*45)
v,f = 29.7 m/s
(g)
KE = (1/2)mv^2
KE = (1/2)*4*29.7^2
KE = 1764.18 J
this actually shows the law conservation of energy, which is ΔPE = -ΔKE
hope this helps~ :)
(a) To calculate the potential energy of an object, we can use the formula:
Potential energy = mass × acceleration due to gravity × height
Given that the mass of the block is 4 kg and assuming the acceleration due to gravity is 9.8 m/s², the potential energy of the block on the floor would be:
Potential energy = 4 kg × 9.8 m/s² × 0 m
Potential energy = 0 Joules
(b) The block is stationary on the floor, so it does not have any kinetic energy. Therefore, the kinetic energy is also 0 Joules.
(c) When the block is raised to a height of 2 m, we can calculate its potential energy using the same formula:
Potential energy = mass × acceleration due to gravity × height
Potential energy = 4 kg × 9.8 m/s² × 2 m
Potential energy = 78.4 Joules
(d) When the block is raised to a height of 4 m, we can again calculate its potential energy:
Potential energy = 4 kg × 9.8 m/s² × 4 m
Potential energy = 156.8 Joules
(e) Raising the block to a height of 45 m, the potential energy is:
Potential energy = 4 kg × 9.8 m/s² × 45 m
Potential energy = 1764 Joules
(f) When the block is dropped to the ground, it will accelerate due to gravity. To find its velocity when it hits the ground, we can use the equation:
Final velocity = √(2 × acceleration due to gravity × height)
Final velocity = √(2 × 9.8 m/s² × 45 m)
Final velocity = √(882 m²/s²)
Final velocity ≈ 29.7 m/s
(g) To calculate the kinetic energy of the block when it hits the ground, we can use the formula:
Kinetic energy = 0.5 × mass × velocity²
Given the mass of the block is 4 kg and the final velocity is approximately 29.7 m/s, the kinetic energy is:
Kinetic energy = 0.5 × 4 kg × (29.7 m/s)²
Kinetic energy = 883.26 Joules
To answer the given questions, we need to understand the concepts of potential energy, kinetic energy, and gravitational potential energy. Here's how you can find the solutions:
(a) The potential energy of the block on the floor can be determined using the equation: Potential Energy = mass x gravitational acceleration x height. In this case, the height is zero since the block is on the floor. Therefore, its potential energy is 0 Joules.
(b) Since the block is not moving (at rest) on the floor, it has no kinetic energy. Kinetic energy is given by the equation: Kinetic Energy = (1/2) x mass x velocity^2.
(c) When the block is raised to a height of 2 meters, we can use the same potential energy formula. Potential Energy = mass x gravitational acceleration x height. Substituting the given values, we get potential energy = 4kg x 9.8 m/s^2 x 2m = 78.4 Joules.
(d) Similarly, when the block is raised to a height of 4 meters, potential energy = 4kg x 9.8 m/s^2 x 4m = 156.8 Joules.
(e) When the block is raised to a height of 45 meters, potential energy = 4kg x 9.8 m/s^2 x 45m = 1764 Joules.
(f) When the block is dropped from a height, it gains kinetic energy as it falls. To find its speed when it hits the ground, we can use the equation: Kinetic Energy = Potential Energy. However, we need to consider that all the potential energy is converted to kinetic energy, so Kinetic Energy = Potential Energy = 1764 Joules. Using the kinetic energy equation, we have 1764 Joules = (1/2) x mass x velocity^2. Plugging in the mass (4kg), we can solve for the velocity. The equation becomes 1764 Joules = (1/2) x 4kg x velocity^2. We can rearrange it to find the velocity: velocity^2 = (2 x 1764 Joules) / 4kg. By calculating this, we find velocity^2 = 882, and therefore velocity ≈ 29.7 m/s when it hits the ground.
(g) The kinetic energy of the block when it hits the ground is the same as the potential energy right before it fell (1764 Joules), as explained in the previous step.