A 5000 kg truck rear ends a 2500 kg car. Both vehicles applied brakes at the moment of the impact and began to skid. After the impact, the acceleration of the truck is -2 m/s and the acceleration of the car is -3 m/s. If the car skid for 30 m and the truck for 2 m, how fast was the truck going after the collision?

To determine the final velocity of the truck after the collision, we'll use the equation:

vf^2 = vi^2 + 2ax

Where:
- vf is the final velocity of the truck
- vi is the initial velocity of the truck
- a is the acceleration of the truck
- x is the distance the truck skidded

Given:
- Initial velocity of the truck (vi) = ?
- Acceleration of the truck (a) = -2 m/s^2
- Distance the truck skidded (x) = 2 m

Plugging in the values, the equation becomes:

vf^2 = vi^2 + 2(-2)(2)
vf^2 = vi^2 - 8

Now, let's find the initial velocity of the truck using the equation:

vf = vi + at

Given:
- Acceleration of the truck (a) = -2 m/s^2
- Time (t) = 1 (since the truck skidded for 2 meters and the truck's acceleration was -2 m/s^2)

Plugging in the values, the equation becomes:

0 = vi + (-2)(1)
0 = vi - 2

Rearranging the equation:

vi = 2

Now, substituting vi = 2 into the equation vf^2 = vi^2 - 8:

vf^2 = 2^2 - 8
vf^2 = 4 - 8
vf^2 = -4

Since we're looking for a positive value, we can ignore the negative sign. Hence, the final velocity of the truck (vf) is:

vf = √4
vf = 2 m/s

Therefore, the speed of the truck after the collision is 2 m/s.

To find the speed of the truck after the collision, we need to use the concept of conservation of momentum. The momentum before the collision should be equal to the momentum after the collision.

The momentum (p) is defined as the product of mass (m) and velocity (v). Mathematically, it can be written as:

p = m * v

For the truck, its momentum before the collision is given by:

p_truck_before = m_truck * v_truck_before

Similarly, the momentum of the car before the collision is:

p_car_before = m_car * v_car_before

The total momentum before the collision can be expressed as the sum of the momenta of the truck and the car:

p_total_before = p_truck_before + p_car_before

After the collision, we know the acceleration of the truck (a_truck) and the car (a_car). We can use these values to find the final velocities of the truck and the car using the following kinematic equation:

v_f = v_i + a * t

Since both vehicles start from rest (initial velocity = 0), we can simplify the equation to:

v_f = a * t

For the truck, the final velocity can be calculated as:

v_truck_after = a_truck * t_truck

Similarly, for the car, the final velocity is:

v_car_after = a_car * t_car

Now, let's calculate the time taken for each vehicle to skid using the formula:

t = sqrt((2 * s) / a)

where s is the distance covered and a is the acceleration.

For the truck, the time taken (t_truck) is:

t_truck = sqrt((2 * 2) / -2) = sqrt(2)

For the car, the time taken (t_car) is:

t_car = sqrt((2 * 30) / -3) = sqrt(20)

Finally, substitute the values into the equations for the final velocities:

v_truck_after = -2 * sqrt(2)
v_car_after = -3 * sqrt(20)

Hence, the speed of the truck after the collision is -2 m/s * sqrt(2).