A car with a mass of 1700 kg is traveling directly northeast (45 degrees between north and east) at a speed of 14m/s (31mph), and collides with a smaller car with a mass of 1300kg that is traveling directly south at a speed of 18m/s (40mph). The two cars stick together during the collision. With what speed and direction does the tangled mess of metal move right after the collision?
To determine the speed and direction of the tangled mess of metal right after the collision, we will use the principle of conservation of momentum.
Step 1: Convert the velocities to vector form.
- The velocity of the first car is given by V1 = 14 m/s at an angle of 45 degrees northeast (45 degrees between north and east).
- The velocity of the second car is given by V2 = 18 m/s at an angle of 180 degrees (directly south).
Step 2: Convert the velocities to their horizontal (x) and vertical (y) components.
For V1:
- V1x = V1 * cos(45°)
- V1y = V1 * sin(45°)
Similarly, for V2:
- V2x = V2 * cos(180°)
- V2y = V2 * sin(180°)
Since V2 is traveling directly south, its x-component will be zero (V2x = 0).
Step 3: Calculate the total momentum before the collision.
The momentum (p) is given by the formula p = m * v, where m is the mass and v is the velocity.
For the first car (m1 = 1700 kg):
- p1 = m1 * V1
For the second car (m2 = 1300 kg):
- p2 = m2 * V2
The total momentum before the collision is given by the sum of the individual momenta:
- p_total = p1 + p2
Step 4: Calculate the total mass and the velocity of the tangled mess of metal after the collision.
Since the two cars stick together after the collision, we can treat them as a single system.
The total mass (m_total) is the sum of the individual masses:
- m_total = m1 + m2
The velocity of the tangled mess of metal after the collision (V_final) is given by the formula:
- V_final = p_total / m_total
Step 5: Calculate the x and y components of V_final.
- V_finalx = V_final * cos(θ_final)
- V_finaly = V_final * sin(θ_final)
Step 6: Convert the x and y components back to magnitude and direction.
- Magnitude: V_final_magnitude = sqrt(V_finalx^2 + V_finaly^2)
- Direction: V_final_direction = atan(V_finaly / V_finalx)
Now, let's calculate the values.
Step 2: Convert the velocities to their x and y components.
For V1:
V1x = 14 m/s * cos(45°) ≈ 9.9 m/s
V1y = 14 m/s * sin(45°) ≈ 9.9 m/s
For V2:
V2x = 18 m/s * cos(180°) = 0 m/s
V2y = 18 m/s * sin(180°) = 0 m/s
Step 3: Calculate the total momentum before the collision.
p1 = 1700 kg * 14 m/s ≈ 23800 kg*m/s
p2 = 1300 kg * 18 m/s ≈ 23400 kg*m/s
p_total = p1 + p2 = 23800 kg*m/s + 23400 kg*m/s ≈ 47200 kg*m/s
Step 4: Calculate the total mass and the velocity of the tangled mess of metal after the collision.
m_total = 1700 kg + 1300 kg = 3000 kg
V_final = p_total / m_total = 47200 kg*m/s / 3000 kg ≈ 15.73 m/s
Step 5: Calculate the x and y components of V_final.
V_finalx = V_final * cos(θ_final) = 15.73 m/s * cos(θ_final)
V_finaly = V_final * sin(θ_final) = 15.73 m/s * sin(θ_final)
Step 6: Convert the x and y components back to magnitude and direction.
Magnitude: V_final_magnitude = sqrt(V_finalx^2 + V_finaly^2)
Direction: V_final_direction = atan(V_finaly / V_finalx)
Since the tangled mess of metal moves together after the collision, both cars will have the same final velocity.
Therefore, the tangled mess of metal moves with a speed of approximately 15.73 m/s (rounded to two decimal places) and in a direction determined by V_final_direction.
To solve this problem, we can use the principle of conservation of momentum. According to this principle, the total momentum before the collision is equal to the total momentum after the collision.
1. First, we need to find the momentum of each car before the collision. The momentum of an object is calculated by multiplying its mass by its velocity.
For the first car (car A) traveling northeast:
Momentum (A) = mass (A) * velocity (A)
Mass (A) = 1700 kg
Velocity (A) = 14 m/s
Momentum (A) = 1700 kg * 14 m/s
For the second car (car B) traveling south:
Momentum (B) = mass (B) * velocity (B)
Mass (B) = 1300 kg
Velocity (B) = -18 m/s (since it's traveling south, we take the negative sign)
Momentum (B) = 1300 kg * (-18 m/s)
2. Next, we need to find the total momentum before the collision. This can be done by adding the individual momenta of both cars.
Total Momentum (Before) = Momentum (A) + Momentum (B)
3. Now, we consider the cars sticking together after the collision. Since they stick together, they will have the same final velocity.
Let's assume the final velocity of the tangled mess of metal after the collision is v.
4. Using the principle of conservation of momentum, we can set up an equation:
Total Momentum (Before) = Total Momentum (After)
(Momentum (A) + Momentum (B)) = Total mass * Final velocity
(1700 kg * 14 m/s) + (1300 kg * (-18 m/s)) = (1700 kg + 1300 kg) * v
5. Simplifying the equation:
23800 kg·m/s - 23400 kg·m/s = 3000 kg * v
400 kg·m/s = 3000 kg * v
6. Solving for v:
v = 400 kg·m/s / 3000 kg
v ≈ 0.133 m/s
7. Finally, we have the speed and direction of the tangled mess of metal after the collision. The speed is approximately 0.133 m/s, and the direction can be determined by considering the initial velocities of both cars (northeast and south). To find the angle, we can use inverse tangent:
Angle = arctan(Vy / Vx)
Where Vy represents the final velocity in the y-direction, and Vx represents the final velocity in the x-direction.
Since the tangled mess of metal will only have a final velocity in the y-direction, we can calculate:
Angle = arctan(0.133 m/s / 14 m/s)
Angle ≈ 0.545 degrees
Therefore, the tangled mess of metal moves with a speed of 0.133 m/s at an angle of approximately 0.545 degrees from the north direction.
do conservation of momentum in NS, and EW directions. Notice the final velocity then will be a vector you can determine the angle and magnitude from.
M1*V1 + M2*V2 = M1*V + M2*V.
1700*14[45o] + 1300*18[270o]=1700V+1300V
Divide both sides by 100:
17*14[45] + 13*18[270] = 17V + 13V.
238[45] + 234[270] = 30V.
168.3+168.3i -234i = 30V.
168.3 - 65.7i = 30V.
180.7[-21.3o] = 30V.
V = 6.0 m/s[-21.3] = 6.0m/s[21.3o] S. of
E.