A 1,600 kg car traveling north at 10.0 m/s crashes into a 1,400 kg car traveling east at 15 m/s at an unexpectedly icy intersection. The cars lock together as they skid on the ice. What is their speed after the crash?
Use conservation of momentum to get the north and east components of the final velocity, V. Then compute the resultant V, which will be the hypotenuse.
3000*Vy = 1600*10 = 16,000 kg m/s
3000*Vx = 1400*15 = 21,000 kg m/s
Vy = 5.333 m/s
Vx = 7.00 m/s
Speed = sqrt[Vx^2 + Vy^2] = 8.8 m/s
(Friction has been assunmed negligible becasue of the the ice.
To find the speed of the cars after the crash, we can use the principle of conservation of momentum. The total momentum before the collision is equal to the total momentum after the collision.
The momentum of an object is given by the product of its mass and velocity. Mathematically, it can be expressed as:
momentum = mass × velocity
Before the collision, the momentum of the first car (car 1) in the north direction is given by:
momentum1 = mass1 × velocity1
momentum1 = 1600 kg × 10.0 m/s
Similarly, the momentum of the second car (car 2) in the east direction is given by:
momentum2 = mass2 × velocity2
momentum2 = 1400 kg × 15 m/s
Since the cars lock together after the crash, their combined mass becomes the sum of their individual masses. Therefore, the total mass after the crash is:
total mass = mass1 + mass2
total mass = 1600 kg + 1400 kg
Now, let's calculate the momentum after the collision. Since the cars lock together, their combined velocity can be represented as V. Therefore:
momentum after the collision = total mass × V
According to the conservation of momentum principle, the momentum before and after the collision must be the same. So:
momentum1 + momentum2 = momentum after the collision
(mass1 × velocity1) + (mass2 × velocity2) = (total mass × V)
Substituting the given values:
(1600 kg × 10.0 m/s) + (1400 kg × 15 m/s) = (total mass × V)
Solving the equation will give us the value of V, which is the speed of the cars after the crash.
To solve this problem, we can use the concepts of conservation of momentum and the Pythagorean theorem.
Step 1: Find the momentum of each car before the crash.
The momentum of an object is defined as the product of its mass and velocity.
Momentum of the first car (car 1): p1 = m1 * v1
= 1600 kg * 10.0 m/s = 16,000 kg*m/s
Momentum of the second car (car 2): p2 = m2 * v2
= 1400 kg * 15.0 m/s = 21,000 kg*m/s
Step 2: Find the total momentum before the crash.
The total momentum before the crash is equal to the sum of the individual momenta of the two cars.
Total momentum before the crash: p_total = p1 + p2
= 16,000 kg*m/s + 21,000 kg*m/s = 37,000 kg*m/s
Step 3: Determine the direction of the total momentum.
Since one car is traveling north and the other is traveling east, the total momentum is a vector sum. We can determine the direction of the total momentum using the Pythagorean theorem.
Let's label the north direction as the y-axis and the east direction as the x-axis. The total momentum in the y-direction is given by the momentum of car 1, p1, and the total momentum in the x-direction is given by the momentum of car 2, p2.
Now, we can use the Pythagorean theorem to find the magnitude of the total momentum, P_total, and its direction.
P_total^2 = p1^2 + p2^2
P_total^2 = (16,000 kg*m/s)^2 + (21,000 kg*m/s)^2
P_total^2 = 256,000,000 kg^2 * m^2/s^2 + 441,000,000 kg^2 * m^2/s^2
P_total^2 = 697,000,000 kg^2 * m^2/s^2
P_total = sqrt(697,000,000 kg^2 * m^2/s^2)
P_total ≈ 26,427.25 kg * m/s
Step 4: Determine the direction of the total momentum.
We can determine the direction using the inverse tangent function (tan^-1). Taking the inverse tangent of the y-component divided by the x-component gives us the direction.
Direction angle = tan^-1(p1 / p2)
Direction angle = tan^-1(16,000 kg*m/s / 21,000 kg*m/s)
Direction angle ≈ 39.81 degrees
Step 5: Calculate the speed of the cars after the crash.
Since the cars lock together and skid on the ice, their masses combine into one. So, we add the masses of car 1 and car 2.
Total mass after the crash: m_total = m1 + m2
= 1600 kg + 1400 kg = 3000 kg
Now, we can calculate the speed of the cars after the crash using the total momentum and total mass.
Speed after the crash: v_total = P_total / m_total
= (26,427.25 kg*m/s) / (3000 kg)
≈ 8.81 m/s
Therefore, the speed of the cars after the crash is approximately 8.81 m/s.