As a van is turning left in an intersection, it crashes into an oncoming car. The van has a mass of 4000 kg and was traveling 55° East of North at 12 km/h before the crash. The car has a mass of 2500 kg and was traveling West at 55 km/h before the crash. After the crash, the van bounces off at 36 km/h to the west (ignore any rotational motion of the car and van). What is the velocity of the car immediately after the crash?
Conserve momentum
4000<36,0> + 2500<vx,vy> = 4000<12sin55°,12cos55°> + 2500<-55,0>
vx = (4000*12sin55° - 2500*55 - 4000*36)/2500
vy = (4000*12cos55°)/2500
All angles are measured CW from +y-axis.
Given: M1 = 4,000kg, V1 = 12km/h[55o] E. of N.
M2 = 2500kg, V2 = 55km/h[270o].
V3 = 36km/h[270o] = Velocity of M1 after crash.
V4 = Velocity of M2 after crash.
Momentum before crash = Momentum after.
M1*V1+M2*V2 = M1*V3+M2*V4
4000*12[55o]+2500*55[270o] = 4000*36[270o]+2500V4
48,000[55o]+137,500[270o] = 144,000[270o]+2500V4
48,000[55o] - 6500[270o] = 2500V46
V4 = 19.2[55o] - 2.6[270o] = (19.2*sin55-2.6*sin270)+(19.2*cos55-2.6*cos270)i
V4 = 18.3+11.0i = 21.4km/h[59o].
To solve this problem, we will need to use the principle of conservation of momentum.
The principle of conservation of momentum states that the total momentum before an interaction is equal to the total momentum after the interaction, as long as no external forces are acting on the system.
Step 1: Convert the given velocities to meters per second (m/s) for consistency.
The initial velocity of the van is 12 km/h. To convert this to m/s, we can use the conversion factor: 1 km/h = 0.28 m/s.
So, the initial velocity of the van is 12 km/h * 0.28 m/s = 3.36 m/s.
The initial velocity of the car is 55 km/h. Converting to m/s, we get: 55 km/h * 0.28 m/s = 15.4 m/s.
Step 2: Calculate the initial momentum of the van and the car separately.
The momentum of an object is calculated by multiplying its mass by its velocity.
The initial momentum of the van = mass of the van * velocity of the van
= 4000 kg * 3.36 m/s
= 13440 kg·m/s
The initial momentum of the car = mass of the car * velocity of the car
= 2500 kg * (-15.4 m/s) (negative since the car is traveling in the opposite direction)
= -38500 kg·m/s
Step 3: Apply the principle of conservation of momentum to find the velocity of the car after the crash.
According to the principle of conservation of momentum, the total momentum before the crash is equal to the total momentum after the crash.
Momentum before the crash = Momentum after the crash
(Initial momentum of the van + Initial momentum of the car) = (Final momentum of the van + Final momentum of the car)
(13440 kg·m/s + (-38500 kg·m/s)) = (Final momentum of the van + Final momentum of the car)
It is given that the final momentum of the van is -36 km/h * 0.28 m/s (to convert to m/s) in the opposite direction, so we can write it as (-36 km/h * 0.28 m/s) = (-10.08 m/s).
Hence, (-38500 kg·m/s + 13440 kg·m/s) = (-10.08 m/s + Final momentum of the car)
Simplifying the equation:
(-25060 kg·m/s) = (-10.08 m/s + Final momentum of the car)
Adding 10.08 m/s to both sides:
Final momentum of the car = (-25060 kg·m/s + 10.08 m/s)
Finally, divide the resultant momentum by the mass of the car to find the final velocity of the car:
Final velocity of the car = Final momentum of the car / mass of the car
Plug in the known values:
Final velocity of the car = [(-25060 kg·m/s + 10.08 m/s) / 2500 kg]
Simplifying the equation gives the velocity of the car immediately after the crash.