# A 1200 kg sedan is traveling at 10m/s when it is hit from behind by a 1000 kg convertible. The bumpers of the cars crush and lock. Their combined speed just after the crash is 16 m/s. What is the velocity of the convertible before the collision? Please show your work.

## momentum is conserved

total momentum = sedan + convertible

(1200 + 1000) 16 =
... (1200 * 10) + 1000 v

## To solve this problem, we can apply the principle of conservation of momentum. The total momentum before the collision is equal to the total momentum after the collision.

Let's denote the velocity of the sedan before the collision as v1 and the velocity of the convertible before the collision as v2.

According to the principle of conservation of momentum:

(mass of sedan x velocity of sedan before the collision) + (mass of convertible x velocity of convertible before the collision)
= (mass of sedan x velocity of sedan after the collision) + (mass of convertible x velocity of convertible after the collision)

Using the given values:

(1200 kg x 10 m/s) + (1000 kg x v2) = (1200 kg + 1000 kg) x 16 m/s

Simplifying the equation:

12000 kg m/s + 1000 kg x v2 = 38000 kg m/s

Subtracting 12000 kg m/s from both sides of the equation:

1000 kg x v2 = 26000 kg m/s

Dividing both sides of the equation by 1000 kg:

v2 = 26 m/s

Therefore, the velocity of the convertible before the collision is 26 m/s.

## To solve this problem, we can use the principle of conservation of linear momentum. The total momentum before the collision is equal to the total momentum after the collision.

The momentum (p) of an object is calculated by multiplying its mass (m) by its velocity (v):

p = m * v

Let's denote the sedan's mass as m1 = 1200 kg, the sedan's velocity before the collision as v1 = 10 m/s, the convertible's mass as m2 = 1000 kg, and the convertible's velocity before the collision as v2 (which we want to find).

The total momentum before the collision is given by:

p_total_before = m1 * v1 + m2 * v2

After the collision, the bumpers of the cars crush and lock, and their combined mass is the sum of their individual masses (m1 + m2 = 1200 kg + 1000 kg = 2200 kg). The remaining unknown value is the combined velocity of the cars after the collision, which is given as v_total_after = 16 m/s.

The total momentum after the collision is then:

p_total_after = (m1 + m2) * v_total_after

According to the conservation of linear momentum principle, the total momentum before and after the collision should be equal:

p_total_before = p_total_after

Therefore, we can set up the equation:

m1 * v1 + m2 * v2 = (m1 + m2) * v_total_after

Plugging in the given values, we have:

(1200 kg * 10 m/s) + (1000 kg * v2) = (1200 kg + 1000 kg) * 16 m/s

Now we can solve for v2:

12000 kg * m/s + 1000 kg * v2 = 2200 kg * 16 m/s

12000 kg * m/s + 1000 kg * v2 = 35200 kg * m/s

1000 kg * v2 = 35200 kg * m/s - 12000 kg * m/s

1000 kg * v2 = 23200 kg * m/s

v2 = (23200 kg * m/s) / 1000 kg

v2 = 23.2 m/s

Therefore, the velocity of the convertible before the collision was 23.2 m/s.