Question 9.56: In the "before" part of Fig. 9-60, car A (mass 1100 kg) is s......
In the “before” part of Fig. 9-60, car A (mass 1100 kg) is stopped at a traffic light when it is rear-ended by car B (mass 1400 kg). Both cars then slide with locked wheels until the frictional force from the slick road (with a low \mu_{k} of 0.13) stops them, at distances d_{A} = 8.2 m and d_{B} = 6.1 m. What are the speeds of (a) car A and (b) car B at the start of the sliding, just after the collision? (c) Assuming that linear momentum is conserved during the collision, find the speed of car B just before the collision. (d) Explain why this assumption may be invalid.
(a) The magnitude of the deceleration of each of the cars is a = f /m = \mu_{k}\, mg/m = μ_kg. If a car stops in distance d, then its speed v just after impact is obtained from Eq. 2-16:
\nu^{2}=\nu_{0}^{2}+2a(x-x_{0}). (2-16)
\nu^{2}=\nu_{0}^{2}+2a d\Rightarrow\nu=\sqrt{2a d}=\sqrt{2\mu_{k}g d}
since \nu_{0} = 0 (this could alternatively have been derived using Eq. 8-31). Thus,
\Delta E_{\mathrm{th}}=f_{k}d. (increase in thermal energy by sliding). (8-31)
\nu_{A}=\sqrt{2\mu_{k}g d_{A}}=\sqrt{2(0.13)(9.8\,\mathrm{{m/s^{2}}})(8.2\,\mathrm{{m)}}}\;=4.6~\mathrm{{m/s}}.
(b) Similarly, \nu_{B}={\sqrt{2\mu_{k}g d_{B}}}={\sqrt{2(0.13)(9.8\,{\mathrm{m/s}}^{2})(6.1~\mathrm{m})}}=3.9~\mathrm{m/s}.
(c) Let the speed of car B be v just before the impact. Conservation of linear momentum gives m_{B}\nu=m_{A}\nu_{A}+m_{B}\nu_{B}, or
\nu={\frac{(m_{A}\nu_{A}+m_{_B}\nu_{B})}{m_{B}}}={\frac{(1100)(4.6)+(1400)(3.9)}{1400}}=7.5~\mathrm{m/s}.
(d) The conservation of linear momentum during the impact depends on the fact that the only significant force (during impact of duration Δt) is the force of contact between the bodies. In this case, that implies that the force of friction exerted by the road on the cars is neglected during the brief Δt. This neglect would introduce some error in the analysis. Related to this is the assumption we are making that the transfer of momentum occurs at one location, that the cars do not slide appreciably during Δt, which is certainly an approximation (though probably a good one). Another source of error is the application of the friction relation Eq. 6-2 for the sliding portion of the problem (after the impact); friction is a complex force that Eq. 6-2 only partially describes.
f_{k}=\mu_{k}F_{N}, (6-2)