A small car of mass 1200 kg traveling east at 60 km/hr collides at an intersection with a truck of mass 3000 kg that is traveling due north at 40 km/hr (
[link] ). The two vehicles are locked together. What is the velocity of the combined wreckage?
A large truck moving north is about to collide with a small car moving east. The final momentum vector has both
x - and
y -components.
Strategy
First off, we need a closed system. The natural system to choose is the (car + truck), but this system is not closed; friction from the road acts on both vehicles. We avoid this problem by restricting the question to finding the velocity at the instant just after the collision, so that friction has not yet had any effect on the system. With that restriction, momentum is conserved for this system.
Since there are two directions involved, we do conservation of momentum twice: once in the
x -direction and once in the
y -direction.
Solution
Before the collision the total momentum is
After the collision, the wreckage has momentum
Since the system is closed, momentum must be conserved, so we have
We have to be careful; the two initial momenta are not parallel. We must add vectorially (
[link] ).
Graphical addition of momentum vectors. Notice that, although the car’s velocity is larger than the truck’s, its momentum is smaller.
If we define the +
x -direction to point east and the +
y -direction to point north, as in the figure, then (conveniently),
Therefore, in the
x -direction:
and in the
y -direction:
Applying the Pythagorean theorem gives
As for its direction, using the angle shown in the figure,
This angle is east of north, or
counterclockwise from the +
x -direction.
Significance
As a practical matter, accident investigators usually work in the “opposite direction”; they measure the distance of skid marks on the road (which gives the stopping distance) and use the work-energy theorem along with conservation of momentum to determine the speeds and directions of the cars prior to the collision. We saw that analysis in an earlier section.
Check Your Understanding Suppose the initial velocities were
not at right angles to each other. How would this change both the physical result and the mathematical analysis of the collision?
Were the initial velocities not at right angles, then one or both of the velocities would have to be expressed in component form. The mathematical analysis of the problem would be slightly more involved, but the physical result would not change.
A common scuba tank is an aluminum cylinder that weighs 31.7 pounds empty (
[link] ). When full of compressed air, the internal pressure is between 2500 and 3000 psi (pounds per square inch). Suppose such a tank, which had been sitting motionless, suddenly explodes into three pieces. The first piece, weighing 10 pounds, shoots off horizontally at 235 miles per hour; the second piece (7 pounds) shoots off at 172 miles per hour, also in the horizontal plane, but at a
angle to the first piece. What is the mass and initial velocity of the third piece? (Do all work, and express your final answer, in SI units.)
