Two particles \(A\) and \(B\) are placed side by side on rough horizontal ground, and are simultaneously projected along the ground in the same direction with speeds \(\quantity{14}{m\,s^{-1}}\) and \(\quantity{21}{m\,s^{-1}}\) respectively. \(A\) comes to rest after travelling \(\quantity{70}{m}\), and \(B\) after travelling \(\quantity{35}{m}\). Calculate the coefficient of friction between each particle and the ground, and show that the times for which the particles are in motion are in the ratio \(3:1\).

Take \(g\) to be \(\quantity{9.8}{m\,s^{-2}}\).

Let’s start by drawing a diagram.

Image showing the two particles A with initial speed 14 m/s travelling 70m to halt and B with initial speed 21m/s travelling 35m to halt

The particles are moving so friction is a maximum and we can use the relationship \(F=\mu R\), where \(R\) is the normal reaction and \(\mu\) is the coefficient of friction. \(F\) is the only horizontal force on the object.

We know \(R=mg\) and Newton’s 3rd law tells us that \(F=ma\) leading to \(\mu=a/g\) (where \(a\) is directed to the left, in the same direction as the frictional force).

To find the coefficient of friction for each particle we must first find its acceleration. For this we can use the equation \(v^2=u^2+2as\).

\(a_A = \dfrac{0-14^2}{2\times70} =\dfrac{-196}{140} = -\dfrac{7}{5}\) so \(\mu_A=\dfrac{7}{5g}=\dfrac{1}{7}\).

\(a_B = \dfrac{0-21^2}{2\times35} =\dfrac{-441}{70} = -\dfrac{63}{10}\) so \(\mu_B=\dfrac{63}{10g}=\dfrac{9}{14}\).

To find out the time of travel we can use \(v=u+at\).

\(t_A = \dfrac{0-14}{-7/5}=10\) and \(t_B = \dfrac{0-21}{-63/10}=\dfrac{10}{3}\).

So the ratio \(t_A:t_B=10:\frac{10}{3}=3:1\) as required.

Show also that \(A\) passes \(B\) before either particle comes to rest, and calculate the speed of each particle when this happens.

Using \(s=ut+\frac{1}{2} at^2\) to find distances at time \(t\), \[\begin{align*} s_A &=14t-\frac{7}{10}t^2 \\ s_B &=21t-\frac{63}{20}t^2. \end{align*}\]

They pass when \(s_A=s_B\), so subtracting the above equations, \(7t-\frac{49}{20}t^2=0\) which means \(t=0\) (i.e. the start) or \(t=\frac{20}{7}\).

At this time \(A\) has speed \(14-\frac{7}{5} \times \frac{20}{7}= \quantity{10}{m\,s^{-1}}\) and \(B\) has speed \(21-\frac{63}{10} \times \frac{20}{7}=\quantity{3}{m\, s^{-1}}\). This is clearly before either particle has come to rest.