A 1400-kg car traveling east at 25m/s collides with a 1800-kg car traveling at a speed of 20m/s in a direction that makes angle of 40degree south of west. The cars stick together after the collision. What is the magnitude and direction of the velocity of the cars after the collision?
momentum conserved. Write momnetum equations in x, y directions.
M1*V1 + M2*V2 = M1*V + M2*V.
1400*25 + 1800*20[180+40]=1400*V+1800*V.
Divide both sides by 100:
14*25 + 18*20[220o] = 14V + 18V.
350 + 360[220o] = 32V.
350 - 276-231i = 32V.
74 - 231i = 32V.
242.6[-72.2o] = 32V.
V = 7.6m/s[-72.2o]=7.6m/s[72o] S. of E.
To solve this problem, we can use the principle of conservation of linear momentum. According to this principle, the total momentum before the collision is equal to the total momentum after the collision.
Step 1: Find the initial momentum of each car.
The momentum of an object is given by the product of its mass and velocity. We can calculate the initial momentum of each car using the following formula:
Momentum = Mass x Velocity
The momentum of the 1400-kg car traveling east at 25m/s is:
Momentum1 = 1400 kg x 25 m/s = 35,000 kg m/s (east)
The momentum of the 1800-kg car traveling at a speed of 20m/s in a direction that makes an angle of 40 degrees south of west can be decomposed into its horizontal and vertical components. The horizontal component can be calculated by multiplying the speed by the cosine of the angle, and the vertical component can be calculated by multiplying the speed by the sine of the angle. These components represent the x and y axis, respectively.
Horizontal component = 20 m/s * cos(40 degrees) ≈ 15.29 m/s
Vertical component = 20 m/s * sin(40 degrees) ≈ -12.90 m/s
The momentum of the 1800-kg car is:
Momentum2 = 1800 kg x (15.29 m/s + 0) ≈ 27,522 kg m/s (east)
(This is because the car is initially traveling south of west, the vertical component contributes to the eastward momentum)
Step 2: Find the total momentum before the collision.
The total momentum before the collision is the sum of the individual momenta of the two cars:
Total Momentum before collision = Momentum1 + Momentum2
Total Momentum before collision = 35,000 kg m/s (east) + 27,522 kg m/s (east) = 62,522 kg m/s (east)
Step 3: Find the final velocity of the combined cars.
The final velocity of the combined cars can be calculated by dividing the total momentum before the collision by the combined mass of the cars. Since the cars stick together after the collision, the total mass is the sum of the individual masses:
Total mass = Mass1 + Mass2
Total mass = 1400 kg + 1800 kg = 3200 kg
Final velocity = Total Momentum before collision / Total mass
Final velocity = 62,522 kg m/s (east) / 3200 kg ≈ 19.54 m/s (east)
Therefore, the magnitude of the velocity of the cars after the collision is approximately 19.54 m/s, and the direction is east.