A BMW of mass \(2.0 \times 10^{3} \mathrm{kg}\) is traveling at $42 \mathrm{m} / \mathrm{s} .\( It approaches a \)1.0 \times 10^{3} \mathrm{kg}$ Volkswagen going \(25 \mathrm{m} / \mathrm{s}\) in the same direction and strikes it in the rear. Neither driver applies the brakes. Neglect the relatively small frictional forces on the cars due to the road and due to air resistance. (a) If the collision slows the BMW down to \(33 \mathrm{m} / \mathrm{s},\) what is the speed of the \(\mathrm{VW}\) after the collision? (b) During the collision, which car exerts a larger force on the other, or are the forces equal in magnitude? Explain.

To determine the speed of the Volkswagen after the collision, we'll use the conservation of linear momentum principle. The total momentum before the collision must equal the total momentum after the collision. We can write this as: \(m_{1} v_{1i} + m_{2} v_{2i} = m_{1} v_{1f} + m_{2} v_{2f}\) where \(m_{1} = 2.0 \times 10^{3} \mathrm{kg}\) (mass of BMW) \(v_{1i} = 42 \mathrm{m/s}\) (initial velocity of BMW) \(m_{2} = 1.0 \times 10^{3} \mathrm{kg}\) (mass of Volkswagen) \(v_{2i} = 25 \mathrm{m/s}\) (initial velocity of Volkswagen) \(v_{1f} = 33 \mathrm{m/s}\) (final velocity of BMW) We need to solve for \(v_{2f}\) (the final velocity of the Volkswagen).

Plugging the given values into the momentum conservation equation, we can rearrange the equation to solve for \(v_{2f}\): \(v_{2f} = \frac{m_{1} v_{1i} + m_{2} v_{2i} - m_{1} v_{1f}}{m_{2}}\) \(v_{2f} = \frac{(2.0 \times 10^{3} \mathrm{kg})(42 \mathrm{m/s}) + (1.0 \times 10^{3} \mathrm{kg})(25 \mathrm{m/s}) - (2.0 \times 10^{3} \mathrm{kg})(33 \mathrm{m/s})}{1.0 \times 10^{3} \mathrm{kg}}\) \(v_{2f} = \frac{84000 \mathrm{kg \thinspace m/s} + 25000 \mathrm{kg \thinspace m/s} - 66000 \mathrm{kg \thinspace m/s}}{1.0 \times 10^{3} \mathrm{kg}}\) \(v_{2f} = \frac{43000 \mathrm{kg \thinspace m/s}}{1.0 \times 10^{3} \mathrm{kg}}\) \(v_{2f} = 43 \mathrm{m/s}\) The final velocity of the Volkswagen after the collision is \(43 \mathrm{m/s}\).

According to Newton's third law, the force exerted by the BMW on the Volkswagen is equal in magnitude but opposite in direction to the force exerted by the Volkswagen on the BMW. Therefore, the forces are equal in magnitude.