At the intersection of Texas Avenue and University Drive, a blue, subcompact car with mass 820 kg traveling east on University collides with a maroon pickup truck with mass 2000 kg that is traveling north on Texas and ran a red light. The two vehicles stick together as a result of the collision, and, after the collision, the wreckage is sliding at 16.0 m/s in the direction 24.0° east of north. Calculate the speed of each vehicle before the collision. The collision occurs during a heavy rainstorm. You can ignore friction forces between the vehicles and the wet road.
East momentum before = 820 c
North momentum before = 2000 t
those will be the same after
mass after = 2820
East after = 2820 * 16 sin 24
North after= 2820 * 16 cos 24
so 820 c = 2820 * sin 24
and 2000 t = 2820 * cos 24
To solve this problem, we can use the principle of conservation of linear momentum. According to this principle, the total momentum before the collision should be equal to the total momentum after the collision.
1. Convert the given angle from degrees to radians:
24.0° × π/180 ≈ 0.4189 radians
2. Since the two vehicles stick together after the collision, their combined mass is equal to the sum of their individual masses:
Total mass (m_total) = m1 + m2 = 820 kg + 2000 kg = 2820 kg
3. We can now calculate the x and y-components of the total momentum before and after the collision:
Before the collision:
Momentum of car in the x-direction (p1x) = m1 * v1
Momentum of car in the y-direction (p1y) = 0
Momentum of truck in the x-direction (p2x) = m2 * v2 * cos(0)
Momentum of truck in the y-direction (p2y) = m2 * v2 * sin(0)
Since the y-components are 0 for both vehicles, the total momentum in the y-direction before and after the collision will also be 0.
After the collision:
Momentum of the wreckage in the x-direction (p_total_x) = m_total * v_final * cos(0.4189)
Momentum of the wreckage in the y-diretion (p_total_y) = m_total * v_final * sin(0.4189)
Again, since the wreckage is sliding 24.0° east of north, the y-component of momentum should not be 0.
4. Calculate the total momentum before the collision:
Before the collision:
Total momentum in the x-direction (p_total_x_before) = p1x + p2x = m1 * v1 + m2 * v2
5. Calculate the total momentum after the collision:
After the collision:
Total momentum in the x-direction (p_total_x_after) = p_total_x = m_total * v_final * cos(0.4189)
6. Equate the total momentum before and after the collision:
m1 * v1 + m2 * v2 = m_total * v_final * cos(0.4189)
7. Calculate the speed of each vehicle before the collision by solving the equation:
v1 = (m_total * v_final * cos(0.4189) - m2 * v2) / m1
8. Substitute the given values:
v1 = (2820 kg * 16.0 m/s * cos(0.4189) - 2000 kg * v2) / 820 kg
Now, to find the value of v2, we need another equation. We'll use the conservation of kinetic energy to solve for v2.
9. Before the collision, the total kinetic energy (KE) is the sum of the kinetic energies of the car and the truck:
KE_before = (1/2) * m1 * v1^2 + (1/2) * m2 * v2^2
10. After the collision, the total kinetic energy is the same but now it includes the wreckage:
KE_after = (1/2) * m_total * v_final^2
11. Equate the total kinetic energies before and after the collision:
(1/2) * m1 * v1^2 + (1/2) * m2 * v2^2 = (1/2) * m_total * v_final^2
12. Solve for v2:
v2^2 = (m_total * v_final^2 - m1 * v1^2) * (2 / m2)
13. Substitute the given values:
v2^2 = (2820 kg * (16.0 m/s)^2 - 820 kg * v1^2) * (2 / 2000 kg)
14. Calculate v2 by taking the square root:
v2 = sqrt((2820 kg * (16.0 m/s)^2 - 820 kg * v1^2) * (2 / 2000 kg))
Now, you can substitute the value of v2 back into the equation for v1 to calculate its value.
Please note that there may be a rounding error due to the use of approximations in the calculations.
To solve this problem, we need to apply the principle of conservation of momentum. According to this principle, the total momentum before the collision should be equal to the total momentum after the collision.
Let's break down the problem into two dimensions, horizontal (East/West) and vertical (North/South), and define our positive directions accordingly.
Given:
- Mass of the blue car (m1) = 820 kg
- Mass of the maroon pickup truck (m2) = 2000 kg
- Velocity of the wreckage after the collision (Vf) = 16.0 m/s
- Angle of wreckage's direction (θ) = 24.0° east of north
Now, let's solve for the speed of each vehicle before the collision:
Step 1: Resolve the wreckage's velocity vector into its horizontal and vertical components.
- Horizontal component (Vf_x) = Vf * cos(θ)
- Vertical component (Vf_y) = Vf * sin(θ)
Step 2: Calculate the total momentum after the collision.
- Momentum of the wreckage (Pf_wreckage) = (m1 + m2) * Vf
- Horizontal component of wreckage's momentum (Pf_wreckage_x) = (m1 + m2) * Vf_x
- Vertical component of wreckage's momentum (Pf_wreckage_y) = (m1 + m2) * Vf_y
Step 3: Apply the conservation of momentum principle to find the total momentum before the collision. Since no external forces are acting, the total momentum remains constant.
- Total momentum before the collision (Pi_total) = Pf_wreckage
- Horizontal component of total momentum before the collision (Pi_total_x) = Pf_wreckage_x
- Vertical component of total momentum before the collision (Pi_total_y) = Pf_wreckage_y
Step 4: Calculate the individual momenta of each vehicle before the collision.
- Momentum of the blue car before the collision (Pi_m1) = m1 * V1 (to be determined)
- Momentum of the maroon pickup truck before the collision (Pi_m2) = m2 * V2 (to be determined)
Step 5: Equate the horizontal and vertical components of the total momentum before the collision to solve for the individual momenta:
- Pi_total_x = Pi_m1 + Pi_m2
- Pi_total_y = 0 (since there is no vertical momentum before the collision)
Step 6: Rearrange the equations and solve for V1 and V2.
- V1 = Pi_m1 / m1
- V2 = Pi_m2 / m2
By solving Steps 5 and 6, we can determine the speeds of each vehicle before the collision.