Question

a collision experiment is conducted using two blocks, block 1 and block 2, that can slide with negligible friction along a horizontal surface. block 1 of mass ( m_1 ) is initially moving with speed ( v_i ) toward block 2, which has a mass ( m_2 ) and is initially at rest, as shown in figure 1. the blocks collide, and after the collision they stick together and move with speed ( v_f ). in multiple trials of this experiment, block 1 is given different initial velocities. data for ( v_f ) as a function of ( v_i ) are recorded, and the data is used to make a best - fit curve, which is a straight line as shown in figure 2. if the experiment were repeated with block 1 replaced by a block of mass ( 2m_1 ), how would the best - fit line to the data change, if at all?
a the slope of the line would be less.
b the slope of the line would be greater.
c the line would become a curve with a slope that is not constant.
d the line would not change.

Explanation:

Step1: Apply conservation of momentum

$$m_1v_i=(m_1 + m_2)v_f$$
$$v_f=\frac{m_1}{m_1 + m_2}v_i$$
The slope of the line \(v_f - v_i\) is \(k_1=\frac{m_1}{m_1 + m_2}\)

Step2: Calculate the new slope

When \(m_1\) is replaced by \(2m_1\)
$$v_f=\frac{2m_1}{2m_1 + m_2}v_i$$
The new slope \(k_2=\frac{2m_1}{2m_1 + m_2}\)

Step3: Compare the slopes

Since \(\frac{2m_1}{2m_1 + m_2}>\frac{m_1}{m_1 + m_2}\) (because \(2m_1(m_1 + m_2)=2m_1^2+2m_1m_2\) and \(m_1(2m_1 + m_2)=2m_1^2+m_1m_2\), and \(2m_1^2+2m_1m_2>2m_1^2+m_1m_2\) for \(m_1>0,m_2>0\))

Answer:

B. The slope of the line would be greater.