Conservation Of Momentum

Calculating Speeds

If a 1000 Kg car crashes with a 3000 kg truck moving at 90º of each other and I know how far of the place of the crash they stopped and at what angle and I also know the sliding friction of the surface they are on, can I calculate the speed of both vehicles?

Yes, you can calculate the speeds of both vehicles before the crash using the information about how far they slid after the crash, the angle at which they stopped, and the coefficient of sliding friction. This process involves applying the principles of work-energy and momentum conservation.

Here’s an outline of how you can calculate the speeds of the car and the truck before the crash:

Step 1: Use Work-Energy Theorem to Calculate the Final Velocity After the Crash

The work-energy theorem states that the work done by friction to stop the vehicles equals their kinetic energy just after the collision.

The work done by friction Wfriction is given by:

W_friction = F_friction . d

where:

• ;; ;;F_friction = µ ;; F_normal ;; is the frictional force,

• ;; ;;d ;; is the distance they slid after the crash,

• ;; ;;µ ;; is the coefficient of sliding friction,

• ;; ;;F_normal = (m_car + m_truck) . g ;; is the normal force (their combined weight).

Since the frictional force is constant, the work done by friction is:

Wfriction = µ . (m_car + m_truck) . g . d

This work equals the kinetic energy of the system just after the collision:

KE_after = 1/2 . (m_car + m_truck) . v_after²

So, equating work done to kinetic energy:

µ ;; . (m_car + m_truck) . g . d = 1/2 . (m_car + m_truck) . v_after²

Cancel out the mass term ;; (mcar + mtruck) , and solve for the velocity v_after:

v_after = √;;;(2 . µ . g . d) ;;

Step 2: Use Conservation of Momentum to Find the Initial Velocities of the Car and Truck

Since the collision happens at 90 degrees, you can use vector addition to apply the conservation of momentum.

The total momentum before the collision must equal the total momentum after the collision. In the case of a right-angle collision:

ptotal = √;;;(p_car² ;; + p_truck² ) ;;

The combined momentum after the collision is:

p_total = (m_car + m_truck) . v_after

So,

√;;(p_car2 ; + p_truck² ) ; = (m_car + m_truck) . v_after

Since ; p = m . v , the momentum of the car is ; p_car = m_car . v_car , and the momentum of the truck is ; p_truck = m_truck . v_truck .

Now you can solve for ; vcar and ; vtruck using the angle of motion after the crash. You can break the momentum into components along the x- and y-axes and use the given angle of motion to solve for each initial velocity.

• pcar is the component of momentum in the x-direction.

• ptruck is the component in the y-direction.

Using trigonometry, you can relate these momenta to the total momentum and the angle:

p_truck/p_car ;  ;= tan (alfa)

Now, knowing the combined momentum from the conservation of momentum equation, you can solve for the initial velocities v_car and ; v_truck

Summary:

1. Use the work-energy theorem to calculate the velocity v_after of the car and truck immediately after the collision.

2. Apply conservation of momentum to relate the initial velocities of the car and truck to the combined velocity and angle of motion after the collision.

3. Use the given angle and momentum components to find the individual speeds of both vehicles before the collision.