At an intersection a car of mass 900kg traveling east collides with a pick up truck of mass 1800kg that is traveling North. Two vehicles stick together as a result of the collision, and after collision, the wreckage is sliding at 16m/sec in the direction 24 degrees East of North. Calculate the speed of each vehicle before collision.
To solve this problem, we can use the principle of conservation of momentum. The total momentum before the collision should be equal to the total momentum after the collision.
Step 1: Assign variables to the unknown quantities.
Let's call the car's initial velocity Vc, and the truck's initial velocity Vt.
Step 2: Write down the initial and final momentum equations.
The initial momentum of the car is given by:
Pc = (mass of the car) x (initial velocity of the car) = 900 kg x Vc
The initial momentum of the truck is given by:
Pt = (mass of the truck) x (initial velocity of the truck) = 1800 kg x Vt
The final momentum after the collision is given by:
Pf = (total mass of the wreckage) x (final velocity of the wreckage)
Step 3: Calculate the total mass of the wreckage.
Since the car and the truck stick together after the collision, the total mass is the sum of their masses:
(total mass of the wreckage) = (mass of the car) + (mass of the truck) = 900 kg + 1800 kg = 2700 kg
Step 4: Calculate the horizontal and vertical components of the final velocity.
Given that the wreckage is sliding at an angle of 24 degrees East of North, we can use trigonometry to find the horizontal and vertical components of the velocity.
Horizontal component of the final velocity:
Vfx = (final velocity of the wreckage) x cos(angle)
Vfx = 16 m/s x cos(24°)
Vertical component of the final velocity:
Vfy = (final velocity of the wreckage) x sin(angle)
Vfy = 16 m/s x sin(24°)
Step 5: Rewrite the final momentum equation using the calculated components.
Using the Pythagorean theorem, we can find the final velocity of the wreckage:
Vf = sqrt((Vfx)^2 + (Vfy)^2)
Step 6: Set up the conservation of momentum equation.
According to the conservation of momentum: Pc + Pt = Pf
Replace the variables with their respective equations:
900 kg x Vc + 1800 kg x Vt = 2700 kg x Vf
Step 7: Solve for Vc.
Rearrange the equation to solve for Vc:
Vc = (2700 kg x Vf - 1800 kg x Vt) / 900 kg
Step 8: Solve for Vt.
Substitute the value of Vc into the equation in step 7 and solve for Vt:
Vt = (2700 kg x Vf - 900 kg x Vc) / 1800 kg
Step 9: Plug in the known values and calculate the velocities.
Substitute the values into the equations and use a calculator to find the values of Vc and Vt.
By following these steps, you should be able to calculate the speeds of each vehicle before the collision.