Two cars initially moving at an angle of 112 degrees towards each otherwill stick together after the collision and travel off as a single unit. The collision is therefore completely inelastic.
The two cars of masses m1=1700kg and m2=2200kg, collisde at an intersection. Before the collision, car 1 was travellig eastward at a speed of v1=38km/h, and car 2 was travelling 22 degrees west of north at a speed of v2=38km/h. After the collision, the two cars stick together and travel off in the direction to be determined. What is the final velocity (magnitude and direction of the two cars sticking together)?
momentum is conserved
find the N-S and E-W components of the momentum of each car
add the components , and find the resultant
Given:
M1 = 1700 kg, V1 = 38km/h.
M2 = 2200 kg, V2 = 38 km/h[112o] CCW.
V3 = Velocity of M1 and M2 after collision.
Momentum before = Momentum after
M1*V1 + M2*V2 = M1*V3 + M2*V3.
1700*38 + 2200*38[112o] = 1700V3 + 2200V3,
64,600 + 83,600[112] = 3900V3,
Divide both sides by 3900:
V3 = 16.56 + 21.44[112] = 16.56 + (-8.03) + 19.88i,
V3 = 8.53 + 19.88i.
V3 = sqrt(8.53^2 + 19.88^2) =
TanA = Y/X = 19.88/8.53, A =
To find the final velocity (magnitude and direction) of the two cars sticking together after the collision, we can use the principle of conservation of linear momentum.
The total momentum before the collision is given by the sum of the individual momenta of the two cars:
p_total_initial = p1_initial + p2_initial
Where,
p_total_initial is the total momentum before the collision,
p1_initial is the momentum of car 1 before the collision,
p2_initial is the momentum of car 2 before the collision.
The momentum of a body is given by the product of its mass and velocity:
p = m * v
Where,
p is the momentum,
m is the mass of the body,
v is the velocity of the body.
Let's calculate the momentum of each car before the collision:
p1_initial = m1 * v1_initial
= 1700 kg * (38 km/h)
p2_initial = m2 * v2_initial
= 2200 kg * (38 km/h)
Now, we need to determine the direction of the vector sum of the momenta. Since both cars are moving at angles relative to each other, we need to decompose the initial velocities into their horizontal and vertical components.
For car 1:
v1_initial_horizontal = v1_initial * cos(angle1)
v1_initial_vertical = v1_initial * sin(angle1)
For car 2:
v2_initial_horizontal = v2_initial * cos(angle2)
v2_initial_vertical = v2_initial * sin(angle2)
Given:
angle1 = 0 degrees (car 1 moving eastward)
angle2 = -22 degrees (car 2 moving 22 degrees west of north)
After the collision, the two cars stick together and travel as a single unit. Let's denote the combined mass of the two cars as M (M = m1 + m2). The final velocity of the combined body is given by:
v_final = p_total / M
Now, we can calculate the horizontal and vertical components of the final velocity of the combined body using the principle of conservation of momentum:
v_final_horizontal = (p1_initial_horizontal + p2_initial_horizontal) / M
v_final_vertical = (p1_initial_vertical + p2_initial_vertical) / M
Finally, we can calculate the magnitude and direction of the final velocity using the Pythagorean theorem and trigonometry:
v_final = sqrt(v_final_horizontal^2 + v_final_vertical^2)
angle_final = arctan(v_final_vertical / v_final_horizontal)
Substituting the given values and performing the calculations will yield the final velocity, both in terms of magnitude and direction.