a 2000kg car traveling east at 24m/sec enters an icy (frictionless) intersection and collides with a 3600kg truck traveling south at 10 m/sec. If they become coupled together in the collision, what is their velocity immediately after impact?

The final momentum is the vector sum of the two initial momenta, which are perpendicular in this case. The eastward component is 48,000 and the southward component is 36,000 kg m/s. Note that you have a 3:4:5 right triangle. Divide the resultant momentum of 60,000 kg m/s by the combined mass for the final speed.

Well, isn't that a slippery situation! When the car and truck collide and become coupled together, we need to consider the conservation of momentum.

The car's momentum before the collision can be found using the equation momentum = mass × velocity. So, the car's momentum is 2000 kg × 24 m/s = 48,000 kg m/s.

Similarly, the truck's momentum before the collision is 3600 kg × 10 m/s = 36,000 kg m/s.

Since momentum must be conserved, the total momentum after the collision should be the same as the total momentum before the collision. Adding up the momenta, we get 48,000 kg m/s + 36,000 kg m/s = 84,000 kg m/s.

Since the car and truck become coupled together, their masses add up to give a total mass of 2000 kg + 3600 kg = 5600 kg.

Now, dividing the total momentum by the total mass, we get 84,000 kg m/s ÷ 5600 kg = 15 m/s.

So, after the impact, the velocity of the car and truck together will be approximately 15 m/s. Let's hope they don't slip and slide too much further!

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.

Momentum (p) is defined as the product of an object's mass (m) and its velocity (v):

p = m * v

The momentum of the car before the collision is given by:

Momentum of car before collision = mass of car * velocity of car

= 2000 kg * 24 m/s

= 48000 kg*m/s (east)

The momentum of the truck before the collision is given by:

Momentum of truck before collision = mass of truck * velocity of truck

= 3600 kg * 10 m/s

= 36000 kg*m/s (south)

To find the total momentum before the collision, we need to consider the direction of motion. Since the car is traveling east and the truck is traveling south, we can consider the car's momentum as positive (eastward) and the truck's momentum as negative (southward).

Total momentum before collision = momentum of car before collision + momentum of truck before collision

= (48000 kg*m/s) east + (-36000 kg*m/s) south

= 48000 kg*m/s east - 36000 kg*m/s south

Now, after the collision, the two vehicles become coupled together. Let's assume their combined mass is M and their velocity is V.

Total momentum after collision = combined mass * velocity after collision

= M * V

According to the conservation of momentum, the total momentum before the collision should be equal to the total momentum after the collision. So, we can equate these two expressions:

Total momentum before collision = Total momentum after collision

48000 kg*m/s east - 36000 kg*m/s south = M * V

To determine the combined velocity (V) after the collision, we need to find the combined mass (M) first. The combined mass is the sum of the masses of the car and the truck:

Combined mass (M) = mass of car + mass of truck

= 2000 kg + 3600 kg

= 5600 kg

Now, let's substitute the values we have into the equation to find the final velocity (V):

48000 kg*m/s east - 36000 kg*m/s south = (5600 kg) * V

To determine the direction of the velocity, we need to calculate the ratio of the magnitudes of eastward to southward velocities:

Ratio of magnitudes = eastward magnitude / southward magnitude

= 48000 kg*m/s / 36000 kg*m/s

= 4/3

Since the ratio is greater than 1, we conclude that the final velocity (V) is directed mainly eastward.

Substituting this ratio into the equation, we get:

48000 kg*m/s east - 36000 kg*m/s south = (5600 kg) * V

4 * 36000 kg*m/s east - 3 * 36000 kg*m/s south = (5600 kg) * V

(4 * 36000 kg*m/s - 3 * 36000 kg*m/s) east = (5600 kg) * V

36000 kg*m/s east = (5600 kg) * V

Now, divide both sides by 5600 kg to solve for V:

36000 kg*m/s east / 5600 kg = V

V ≈ 6.429 m/s east

Therefore, the velocity immediately after the impact is approximately 6.429 m/s east.

To find the velocity immediately after the impact, we need to use the principle of conservation of momentum. According to this principle, the total momentum before the collision is equal to the total momentum after the collision.

Momentum is defined as the product of mass and velocity (momentum = mass x velocity).

Let's denote the velocity of the car after the collision as Vc and the velocity of the truck after the collision as Vt.

The momentum of the car before the collision is given by:

Momentum_car_before = mass_car x velocity_car
= 2000 kg x 24 m/s
= 48000 kg m/s (east)

The momentum of the truck before the collision is given by:

Momentum_truck_before = mass_truck x velocity_truck
= 3600 kg x (-10 m/s) [negative because it is traveling south]
= -36000 kg m/s (south)

The total momentum before the collision is the sum of the momenta of the car and the truck:

Total_momentum_before = Momentum_car_before + Momentum_truck_before

Now, since they become coupled together after the collision, the total momentum after the collision will be the same as the total momentum before the collision. So:

Total_momentum_after = Total_momentum_before

Total_momentum_after = (2000 kg + 3600 kg) x Velocity_after_collision

Simplifying the equation:

(5600 kg) x Velocity_after_collision = 48000 kg m/s (east) - 36000 kg m/s (south)

Now let's break down the velocities into their respective components by using vectors. The car's initial velocity is purely eastward, so it has no southward component. The truck's initial velocity is purely southward, so it has no eastward component. Therefore:

Total_momentum_after = (5600 kg) x Velocity_after_collision
Total_momentum_after = (48000 kg m/s east) + (36000 kg m/s south)

To combine the magnitudes and directions, we will use the Pythagorean theorem. The magnitude is the square root of the sum of the squares of the magnitudes, and the direction is given by the inverse tangent:

Total_momentum_after = sqrt((48000^2 + 36000^2) kg m/s)
= sqrt(2304000000 + 1296000000) kg m/s
= sqrt(3600000000) kg m/s
= 60000 kg m/s

Now, we have the total momentum after the collision. To find the velocity, we divide by the total mass:

Velocity_after_collision = Total_momentum_after / Total_mass

Total_mass = mass_car + mass_truck
= 2000 kg + 3600 kg
= 5600 kg

Velocity_after_collision = 60000 kg m/s ÷ 5600 kg
= 10.714 m/s

So, the velocity of the car and the truck immediately after the collision is approximately 10.714 m/s in the direction determined by the components we calculated earlier.