Question

1. using the graphs as a guide, select each statement that correctly describes the relationships between kinetic energy and mass and kinetic energy and speed (velocity). select all that apply. a. the relationship between velocity and kinetic energy is exponential. b. the relationship between mass and kinetic energy is linear. c. the relationship between velocity and kinetic energy is linear. d. when velocity doubles, kinetic energy quadruples. e. when mass increases by half, kinetic energy increases by half.

Explanation:

Step1: Recall the formula for kinetic energy

The formula for kinetic energy is \(E_{k}=\frac{1}{2}mv^{2}\), where \(m\) is mass and \(v\) is velocity.

Step2: Analyze the relationship between mass and kinetic energy

From \(E_{k}=\frac{1}{2}mv^{2}\), if \(v\) is constant, \(E_{k}\) is directly proportional to \(m\) (\(E_{k}=km\), \(k = \frac{1}{2}v^{2}\) (constant)). A direct - proportional relationship is linear. So, when looking at the \(E_{k}-m\) graph (assuming constant \(v\)), the relationship is linear.

Step3: Analyze the relationship between velocity and kinetic energy

From \(E_{k}=\frac{1}{2}mv^{2}\), if \(m\) is constant, \(E_{k}\) is a quadratic function of \(v\) (\(E_{k}=kv^{2}\), \(k=\frac{1}{2}m\) (constant)). A quadratic function \(y = ax^{2}+bx + c\) (\(b = 0,c = 0\) in this case) is an exponential - like (parabolic, which is a type of non - linear, and for positive \(x\) values, it has an exponential - growth - like shape in terms of non - linearity) relationship. If \(v\) doubles (\(v_{2}=2v_{1}\)), then \(E_{k2}=\frac{1}{2}m(2v_{1})^{2}=4\times\frac{1}{2}mv_{1}^{2}=4E_{k1}\).

Answer:

A. The relationship between velocity and kinetic energy is exponential.
B. The relationship between mass and kinetic energy is linear.
D. When velocity doubles, kinetic energy quadruples.