All of these questions can be answered using one of these concepts:
Initial Energy = Final Energy
Initial KE + Initial PE = Final KE + Final PE
where KE is kinetic energy and PE is potential energy.
KE = 1/2mv^2 and PE = mgh
#1 We are only given the initial height of the bobsled, which is h = 150m. We would assume (since not given) that we are starting from rest at the top —> this means our initial velocity is 0. We also know that we can use gravity g = 9.8 m/s^2.
Initially, we are at the top. Let’s make an expression for our initial energy
Initial Energy = initial KE + initial PE
If we are not moving at the top, our KE is 0. We only have PE at the top. Even though we don’t have the mass of the bobsled, we can still make an expression for the energy:
Initial PE = mgh = m(9.8)(150) = 1470m
We will just carry the m with us for now since we don’t have a value for it.
Our Initial Energy = Initial KE + Initial PE = 0 + 1470m = 1470m
Now let’s make an expression for the final energy, or the energy at the bottom of the hill. At the bottom of the hill, our height is now 0 —> h=0. This means we won’t have any potential energy at the bottom. All of the potential that we had at the top of the hill has turned into kinetic energy at the bottom of the hill.
Final Energy = KE final + PE final
PE final = 0
KE final = 1/2mv^2
Final Energy = 1/2mv^2 + 0 = 1/2mv^2
We are asked to find the speed at the bottom of the hill, so now we will use conservation of energy:
Initial Energy = Final Energy
1470m = 1/2mv^2
Even though we didn’t know the mass, we can see that it shows up on both sides of the equation. This means we can cancel it out.
1470 = 1/2v^2
Now we can solve for v
1470 = 1/2v^2 times by 2 on both sides
2940 = v^2 square root on both aides
V = 54.22 m/s at the bottom of the hill = A
For the next problems I will do less of an explanation now that I have explained the general concept in #1
#2 I assume that the fish was initially at rest and decided to work to change its energy.
Change in Kinetic Energy = final KE - initial KE
Initial KE = 0 J
Final KE = 1/2mv^2 where m = 1.4kg and v = 15 m/s
KE = 1/2(1.4)(15^2) = 157.5 J
Change in KE = 157.5 - 0 = 157.5 J = A
#3 The piano is being held at a height above the ground. This means it’s KE is 0 (not falling) and it only has potential energy.
PE = mgh
Mass = 500 kg = m and h = 0.5 m above the ground
we also know g = 9.8 m/s^2
PE = mgh = (500)(9.8)(0.5) = 2450J = A
#4 mass of raindrop m = 0.0005 kg and the height h = 250m. We also know gravity g = 9.8 m/s^2.
It is dropping from a height so it starts with no initial KE and only has Initial PE.
Initial Energy = Initial PE + Initial KE = mgh + 0
Initial Energy = (0.0005)(9.8)(250) = 1.225 J
Because of conservation of energy, Initial Energy = Final Energy. The energy right before the raindrop hits the ground is the same as the energy it has at the top of it’s fall.
At the top, the raindrop had only PE. At the bottom, the raindrop has only KE.
Final Energy = KE final + PE final
Final Energy = KE final + 0
Final Energy = KE final
Using Conservation of energy,
Initial Energy = Final Energy, we know the final energy is also = 1.225 J because it must match the initial. We also know all of our energy at the bottom is only KE. This means all of our final energy is KE, so KE is also = 1.225 J
Answer = 1.225J = A
