Question

Tyana is using lead balls to perform investigations of kinetic energy. She records her observations in two tables.
 

INVESTIGATION 1:  constant mass
Trial Velocity (meters/second) Mass (kilograms) Kinetic Energy (joules)
1 8 3 96
2 2 3 6
3 4 3 24
4 6 3 54
 

INVESTIGATION 2: constant velocity
Trial Velocity (meters/second) Mass (kilograms) Kinetic Energy (joules)
1 2 6 12
2 2 2 4
3 2 5 10
4 2 8 16

Using the patterns observed in Tyana's data, which two graphs best represent the relationships between velocity, mass, and kinetic energy?

A.
image

B.
image

C.
image

D.
image

Answer 1

To answer your question effectively, we first need to analyze the data provided in both investigations to understand the relationships between velocity, mass, and kinetic energy.

### Investigation 1: Constant Mass
Using the formula for kinetic energy (KE), which is:
\[ KE = \frac{1}{2} m v^2 \]
Where:
- \( KE \) = kinetic energy in joules
- \( m \) = mass in kilograms
- \( v \) = velocity in meters/second

For the constant mass of 3 kg in Investigation 1:
1. Trial 1: \( KE = \frac{1}{2} (3)(8^2) = 96 \) joules
2. Trial 2: \( KE = \frac{1}{2} (3)(2^2) = 6 \) joules
3. Trial 3: \( KE = \frac{1}{2} (3)(4^2) = 24 \) joules
4. Trial 4: \( KE = \frac{1}{2} (3)(6^2) = 54 \) joules

The kinetic energy increases with the square of velocity, meaning it shows a parabolic relationship vs. velocity.

### Investigation 2: Constant Velocity
Here, the velocity remains constant (2 m/s), while mass varies:
1. Trial 1: For 6 kg, \( KE = \frac{1}{2} (6)(2^2) = 12 \) joules
2. Trial 2: For 2 kg, \( KE = \frac{1}{2} (2)(2^2) = 4 \) joules
3. Trial 3: For 5 kg, \( KE = \frac{1}{2} (5)(2^2) = 10 \) joules
4. Trial 4: For 8 kg, \( KE = \frac{1}{2} (8)(2^2) = 16 \) joules

In this case, kinetic energy increases linearly with mass since the velocity is constant.

### Summary of Relationships:
- In the first investigation, kinetic energy is a function of the square of velocity (quadratic relationship).
- In the second investigation, kinetic energy is a linear function of mass.

### Graph Selection:
To select the appropriate graphs, you would be looking for:
1. A parabola for Investigation 1 (KE vs. Velocity) showing KE increases quadratically with velocity.
2. A straight line for Investigation 2 (KE vs. Mass) indicating KE increases linearly as mass increases.

Without seeing the actual graphs A, B, C, and D, I cannot definitively choose which two best represent these relationships, but these principles should help you identify them based on the characteristics we discussed. Look for one graph that has a parabolic shape for the KE vs. Velocity (Investigation 1) and another that has a straight line for the KE vs. Mass (Investigation 2).