Question

the data table below compares the speed of the baseball after contact when the bats mass and speed is changed. use this information to answer questions 1 - 3.
what has the greatest effect on the amount of kinetic energy a moving object has?
changes in bats mass:
bat mass (g) | speed of ball after contact (mph)
570 | 68
710 | 73
850 | 76
990 | 78
1140 | 80
changes in bats speed:
bat swing speed (mph) | speed of ball after contact (mph)
20 | 60
27 | 68
34 | 76
41 | 84
48 | 92

Explanation:

Step-by-Step Format:

Step1: Recall the formula for kinetic energy

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

Step2: Analyze the effect of mass and speed on kinetic energy

• Let's assume two cases for mass: If mass \(m_1 = m\) and \(m_2=2m\). Then \(KE_1=\frac{1}{2}mv^{2}\) and \(KE_2=\frac{1}{2}(2m)v^{2}=mv^{2}\), so \(KE_2 = 2KE_1\) (a linear relationship with mass).
• For speed: If \(v_1 = v\) and \(v_2 = 2v\). Then \(KE_1=\frac{1}{2}mv^{2}\) and \(KE_2=\frac{1}{2}m(2v)^{2}= 2mv^{2}\), so \(KE_2=4KE_1\) (a quadratic relationship with speed).

Step3: Check the data in the tables

• In the "Changes in bat's mass" table: When the bat mass changes from \(570g\) to \(1140g\) (doubling approximately, \(1140\div570 = 2\)), the ball speed changes from \(68mph\) to \(80mph\) (an increase of \(\frac{80 - 68}{68}\approx0.176\)).
• In the "Changes in bat's speed" table: When the bat speed changes from \(20mph\) to \(48mph\) (an increase of \(2.4\) times), the ball speed changes from \(60mph\) to \(92mph\) (an increase of \(\frac{92 - 60}{60}\approx0.533\)). The quadratic nature of the speed - kinetic energy relationship (from the formula \(KE=\frac{1}{2}mv^{2}\)) means that small changes in speed lead to relatively larger changes in kinetic energy compared to similar - magnitude changes in mass.

Answer:

The speed of the ball (since kinetic energy \(KE=\frac{1}{2}mv^{2}\) and speed has a quadratic relationship with kinetic energy, while mass has a linear relationship, so speed has a greater effect on kinetic energy).