Math Snap

STEP 1

What is this asking?
We're exploring energy changes, from friction's work on a car, factors influencing gravitational potential energy, how stretching a spring affects its energy, and comparing the kinetic energies of a bowling ball and a bocce ball.
Watch out!
Don't mix up kinetic and potential energy, and remember work can decrease kinetic energy!
Also, pay close attention to how different factors contribute to different types of energy.

STEP 2

1. Friction and Kinetic Energy
2. Gravitational Potential Energy Factors
3. Elastic Potential Energy and Displacement
4. Kinetic Energy Comparison

STEP 3

Friction does work on the car, and work is a transfer of energy.
Since the work done by friction is negative ($-400$ J), this means energy is being taken away from the car's motion.

STEP 4

The car's kinetic energy, its energy of motion, will therefore decrease by the amount of work done by friction.
So, the kinetic energy decreases by $400$ J.

STEP 5

Gravitational potential energy depends on an object's mass, its height relative to a reference point, and the acceleration due to gravity.
It's all about the potential for gravity to do work on the object.

STEP 6

Speed, however, is related to kinetic energy, not potential energy.
So, an object's speed does not affect its gravitational potential energy.

STEP 7

The elastic potential energy stored in a spring is given by $U_s = \frac{1}{2}kx^2$, where $k$ is the spring constant and $x$ is the displacement from equilibrium.

STEP 8

If we double the displacement, the new displacement is $2x$.
Plugging this into the formula, we get $U_s' = \frac{1}{2}k(2x)^2 = \frac{1}{2}k \cdot 4x^2 = 4(\frac{1}{2}kx^2) = 4U_s$.

STEP 9

So, doubling the displacement increases the elastic potential energy by a factor of four!

STEP 10

The kinetic energy of an object is given by $KE = \frac{1}{2}mv^2$, where $m$ is the mass and $v$ is the velocity.

STEP 11

For the bowling ball: $KE = \frac{1}{2}(4.0 \text{ kg})(1.0 \text{ m/s})^2 = \frac{1}{2} \cdot 4.0 \cdot 1.0 = 2.0 \text{ J}$.

STEP 12

For the bocce ball: $KE = \frac{1}{2}(1.0 \text{ kg})(4.0 \text{ m/s})^2 = \frac{1}{2} \cdot 1.0 \cdot 16.0 = 8.0 \text{ J}$.

STEP 13

The bocce ball has more kinetic energy ($8.0$ J) than the bowling ball ($2.0$ J).

6. b. The kinetic energy decreases by $400$ J.
7. d. an object's speed
8. c. The elastic potential energy increases or decreases by a factor of 4.
9. The bocce ball has more kinetic energy ($8.0$ J) than the bowling ball ($2.0$ J).