Question

The figures below show bird's-eye views of six automobile crashes an instant before they occur. The automobiles have different masses and incoming velocities as shown. After impact, the automobiles remain joined together and skid to rest in the direction shown by v_final. Rank these crashes according to the angle θ , measured counterclockwise as shown, at which the wreckage initially skids.

 

Answer 1

The required concepts to solve the problem are collision and momentum conservation.

First, use the momentum conservation to find a general equation for the angle made by the automobiles after the collision.

Then rank, from largest to smallest, the crashes according to the angle measured counterclockwise.

 

Velocity is a vector with both magnitude and direction. The direction of the vector depends on the angle made by the vector with respect to the horizontal or with respect to the vertical. If the angle is made with respect to the horizontal, then the x component of the vector is given by the cosθ component of the vector and the y component of the vector is given by the sinθ component of the vector.

According to the conservation of momentum, the momentum before the collision is equal to the momentum after the collision.

If the collision between two objects is considered, the conservation of momentum gives the relation,

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

Here, m₁ is the mass of the first object, m₂ is the mass of the second object, u₁ is the initial velocity of the first object, u₂ is the initial velocity of the second object, v₁ is the velocity after the collision of the first object and v₂ is the final velocity after the collision of the second object.

Along the x direction, applying conservation of momentum,

Here, m_x is the mass of the car traveling in the x direction, m_y is the mass of the car travelling in the y direction, u_x is the velocity of the car moving in the x direction, θ is the angle made by the combined mass with respect to the horizontal and v_final is the velocity of the combined mass.

Along the direction, applying conservation of momentum,

Here, u_y is the velocity of the car moving in the y direction.

Finding the ratio of the components of the final velocity, we get

Thus, the equation for the angle made by the combined mass after collision is,

The equation for the angle made by the final velocity of the combined mass is,

For the first case, substitute 700kg for m_y, 1200kg for m_x, 15 m/s for u_y, and, 9 m/s for u_x to find θ₁.

Here, θ₁ is the angle in the first case.

For the second case, substitute 1000kg for m_y, 1000kg for m_x, 10 m/s for u_y, and 7 m/s for u_x to find θ₂.

Here, θ₂ is the angle in the second case.

For the third case, substitute 700kg for m_y, 1000kg for m_x, 7 m/s for u_y, and 7 m/s for u_x to find θ₃.

Here, θ₃ is the angle in the third case.

For the fourth case, substitute 1000kg for m_y, 1000kg for m_x, 7 m/s for u_y, and 10 m/s for u_x to find θ₄.

Here, θ₄ is the angle in the fourth case.

For the fifth case, substitute 1500kg for m_y, 1000kg for m_x, 10 m/s for u_y, and 15 m/s for u_x to find θ₅.

Here, θ₆ is the angle in the fifth case.

For the sixth case, substitute 1000kg for m_y, 1000kg for m_x, 10 m/s for u_y, and 10 m/s for u_x to find θ₆.

Here, θ₆ is the angle in the sixth case.

Ans:

The ranking of the crashes from largest to smallest is,