At a busy intersection a 1460-kg car traveling west with a speed of 12 m/s collides head-on with a minivan traveling east with a speed of 9.4 m/s. The cars stick together and move with an initial velocity of 1.5 m/s to the east after the collision.
What is the mass of the minivan?
To determine the mass of the minivan, we need to use the principle of conservation of momentum.
The total momentum before the collision is equal to the total momentum after the collision.
The momentum of an object is calculated by multiplying its mass by its velocity.
Let's denote the mass of the minivan as m_minivan.
Given:
Mass of the car (m_car) = 1460 kg
Initial velocity of the car (v_car_initial) = 12 m/s
Initial velocity of the minivan (v_minivan_initial) = 9.4 m/s
Final velocity of the combined vehicles (v_final) = 1.5 m/s
Using the principle of conservation of momentum:
(m_car * v_car_initial) + (m_minivan * v_minivan_initial) = (m_car + m_minivan) * v_final
Substituting the given values into the equation:
(1460 kg * 12 m/s) + (m_minivan * 9.4 m/s) = (1460 kg + m_minivan) * 1.5 m/s
Simplifying the equation:
17520 kg m/s + 9.4 m/s * m_minivan = 1.5 m/s * (1460 kg + m_minivan)
Multiplying out the terms:
17520 + 9.4 m_minivan = 2190 + 1.5 m_minivan
Rearranging the equation to solve for m_minivan:
9.4 m_minivan - 1.5 m_minivan = 2190 - 17520
7.9 m_minivan = -15330
m_minivan = -15330 / 7.9 kg
Since mass cannot be negative, the mass of the minivan is zero.
To find the mass of the minivan, we can use the law of conservation of momentum. According to this law, the total momentum before the collision is equal to the total momentum after the collision.
The momentum of an object is given by the product of its mass and velocity. Let's assume the mass of the minivan is m kg.
Before the collision:
Momentum of the car = mass of the car × velocity of the car = 1460 kg × 12 m/s = 17520 kg·m/s (to the west)
Momentum of the minivan = mass of the minivan × velocity of the minivan = m kg × (-9.4 m/s) (since the minivan is traveling east, we take the velocity as negative) = -9.4m kg·m/s
After the collision:
Momentum of the combined vehicles = total mass of the vehicles × resulting velocity
The total mass of the vehicles is the sum of the car's mass and the minivan's mass: 1460 kg + m kg = (1460 + m) kg
The resulting velocity is 1.5 m/s to the east.
Using the law of conservation of momentum:
17520 kg·m/s - 9.4m kg·m/s = (1460 + m) kg × 1.5 m/s
Simplifying the equation:
17520 kg·m/s - 9.4m kg·m/s = 2190 kg·m/s + 1.5m kg·m/s
Combining like terms:
(17520 - 2190) kg·m/s = 9.4m kg·m/s + 1.5m kg·m/s
Simplifying further:
15330 kg·m/s = 10.9m kg·m/s
Dividing both sides of the equation by 10.9 kg·m/s:
m = 15330 kg·m/s ÷ 10.9 kg·m/s
Calculating the mass of the minivan:
m ≈ 1409.17 kg
Therefore, the mass of the minivan is approximately 1409.17 kg.
1940
conservation of momentum
1460*12W +M*12E=(M+1460)1.5E
since W=-E, then
-1460*12 +M*9.4 =(M+1460)1.5
-1460(9.4+1.5)=-12M
solve for M, the mass of the minivan