The problem describes a perfectly inelastic collision between a 1000-kg car moving eastward at 20 m/s and a 1500-kg van traveling northward at 30 m/s. In a perfectly inelastic collision, the two objects stick together after the collision and move as one object.
To solve this problem, we can use the principle of conservation of momentum, which states that the total momentum of a closed system remains constant if no external forces act on it.
Let’s define the eastward direction as the positive x-axis and the northward direction as the positive y-axis. The initial momentum of the car is:
pcar = mcar * vcar = (1000 kg) * (20 m/s) * i^ = 20000 kg*m/s * i^
The initial momentum of the van is:
pvan = mvan * vvan = (1500 kg) * (30 m/s) * j^ = 45000 kg*m/s * j^
The total initial momentum of the system is:
pinitial = pcar + pvan = 20000 kgm/s * i^ + 45000 kgm/s * j^
Since the collision is perfectly inelastic, the two objects stick together after the collision and move as one object. Let’s call the final velocity of the combined object as v. The final momentum of the system is:
pfinal = (mcar + mvan) * v = (1000 kg + 1500 kg) * v = 2500 kg * v
According to the principle of conservation of momentum, the total initial momentum of the system is equal to the total final momentum of the system:
pinitial = pfinal
Substituting the values of pinitial and pfinal, we get:
20000 kgm/s * i^ + 45000 kgm/s * j^ = 2500 kg * v * u
where u is the unit vector in the direction of the final velocity. Since the two objects stick together after the collision, the final velocity vector is in the direction of the resultant momentum vector. Therefore, we can write:
u = (20000 kgm/s * i^ + 45000 kgm/s * j^) / sqrt((20000 kgm/s)^2 + (45000 kgm/s)^2)
Simplifying, we get:
u = (4 * sqrt(1301) / 1301) * i^ + (9 * sqrt(1301) / 1301) * j^
Therefore, the final velocity of the combined object is:
v = (20000 kgm/s + 45000 kgm/s) / 2500 kg = 26 m/s
The direction of the final velocity is:
θ = tan^-1(9 * sqrt(1301) / 4 * sqrt(1301)) = 67.4 degrees north of east.
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