We can use conservation of energy if we know all the energy forms.
Friction and inelastic collisions convert kinetic energy to (invisible) thermal energy.

Forces can do work, changing energy. We introduce two kinds of energy:

Gravitational Potential Energy: If you lift a rock from the bottom to the top of a cliff, the work you do equals the increase in gravitational potential energy.

$LaTeX: \Delta E_{gp} = mg\Delta y$ , where

is the acceleration of gravity ~ 10 m/s² (at sea level on earth) and
$LaTeX: \Delta y$ is the height the rock is lifted.

Exercise 1.6.1

You have two rocks: rock A and Rock B. Rock A has twice the mass of rock B, and is also twice as high off the ground as rock B. What is the ratio of the energy of rock A to rock B? Show your work.

Kinetic energy: if you push something causing it to speed up, the work you do equals the increase in kinetic energy.

$LaTeX: E_K = \frac{1}{2}mv^2$

Exercise 1.6.2

You have two cars at the same height: Car A has twice the mass of car B, and is also moving at twice the speed of car B. What is the ratio of the energy of car A to car B? Show your work.

Because energy is conserved, these formulas make it easy to find the speed of a falling object as a function of how high it is: the loss of potential energy is the same as the gain in kinetic energy and vice versa. This is easy if there are no other energy transformations. In real life, friction often turns some of the mechanical energy into thermal energy, and we must account for the heat loss by subtracting it from the mechanical energy.

Exercise 1.6.3

You have two rocks dropped from the same height. Rock A has twice the mass of rock B. What is the ratio of their speeds when they hit the ground?

Exercise 1.6.4

You have two rocks of equal mass. Rock A is dropped from twice the height as rock B. What is the ratio of their speeds when they hit the ground? Please make a drawing and show your work.

Exercise 1.6.5

You go on a super slippery water slide that drops 10 meters to the ground.

What is your speed at the bottom of the slide? Solution below
What is your speed half way down the slide? Help below
What if you are a fast runner and enter the top of the slide at an initial speed of 10 m/s? What would your speed be at the bottom? Help below

Help with B. At first glance, we might be tempted to answer that half way down, we have half our final velocity and conclude the speed at 5 m height is ~ 7 m/s. However, it’s our kinetic energy that is half of what it is at the end. Note that kinetic energy isn't proportional to the velocity, but rather to the square of the velocity, so velocity will increase like the square root of the height. If you cut the kinetic energy in half, you only reduce the speed by $LaTeX: \sqrt{2}$ . So, you gain more than half the final speed by the time you’ve dropped half the height. How can this happen? Consider kinematics that you’re moving slower in the beginning, so you accelerate for a longer time than you do in the second half at higher speed.

Help with C. You might first be tempted to answer $LaTeX: v_f = v_i + \Delta v$ = 24.1 m/s. However, speed is not conserved; energy is conserved. Now, the initial energy is both potential energy and kinetic energy. What final speed do you get? Again, as in B. above, if we start with an initial speed, should the speed we gain during the fall be more or less than 14.1 m/s, why?

The unit of energy = the unit of work = the Joule after James Joule. J = kg(m²/s²).

Exercise 1.6.6

Verify that the Joule is the correct unit for both kinetic and potential energy.

Exercise 1.6.7

I throw a rock upward with an initial speed of 20 m/s.

How high does it get?
What is its speed when it is 10 m high?

Exercise 1.6.8

A 1000 kg car moving at 20 m/s strikes a stationary 4000 kg truck. The two stick together and continue moving on the very slippery ground. Was any thermal energy produced in this collision? If so how many Joules of thermal energy was produced? If not, how can you be sure no thermal energy was produced?

Solution:

Handwritten solution to exercise 1.6.8