Question

Consider the following two-car accident: Two cars of equal mass m collide at an intersection. Driver E was traveling eastward, and driver N, northward. After the collision, the two cars remain joined together and slide, with locked wheels, before coming to rest. Police on the scene measure the length d of the skid marks to be 9 meters. The coefficient of friction μ between the locked wheels and the road is equal to 0.9. (Figure 1) Each driver claims that his speed was less than 14 meters per second (about 31 mph). A third driver, who was traveling closely behind driver E prior to the collision, supports driver E's claim by asserting that driver E's speed could not have been greater than 12 meters per second. Take the following steps to decide whether driver N's statement is consistent with the third driver's contention. Part A

Let the speeds of drivers E and N prior to the collision be denoted by ve and vn, respectively. Find v2, the square of the speed of the two-car system the instant after the collision.
Express your answer terms of ve and vn.

Answer 1

To calculate the square of the speed of the two-car system, v^2, immediately after the collision, we apply the conservation of momentum in both eastward and northward directions and use the Pythagorean theorem, resulting in v^2 being equal to ve^2 + vn^2.

Step-by-step explanation:

To find v2, the square of the speed of the two-car system immediately after the collision, we can apply the principle of conservation of momentum in two dimensions, since momentum is conserved in both the x-direction (eastward) and y-direction (northward). Given that the cars have equal mass m and their speeds before the collision are ve and vn respectively, we have:

• In the eastward direction: momentum before the collision was m × ve.
• In the northward direction: momentum before the collision was m × vn.

After the collision, the cars stick together, moving as a single object of mass 2m. The square of the velocity of this combined mass immediately after the collision, v2, can be found using the Pythagorean theorem, as the velocities in the x and y directions are perpendicular:

v2 = ve2 + vn2