A 1800 kg mass car is traveling south at a velocity of 15 m/s through an intersection. It is hit by a 3500 kg mass car traveling east at 20 m/s. If the two cars collide and become one object (perfectly inelastic). What will be the final displacement of the cars after the collision if the coefficient of kinetic friction between the road and the tires is 0.8?
To determine the final displacement of the cars after the collision, we first need to calculate the momentum of each car before the collision, then use the principle of conservation of momentum.
Momentum (p) is calculated by multiplying an object's mass (m) with its velocity (v). The equation for momentum is:
p = m * v
For car 1 (1800 kg mass car traveling south at 15 m/s), the momentum is calculated as:
p1 = 1800 kg * 15 m/s
For car 2 (3500 kg mass car traveling east at 20 m/s), the momentum is:
p2 = 3500 kg * 20 m/s
To create a unified system, we need to determine the direction of each car's momentum. Let's consider the south direction as positive and the east direction as negative. Therefore, the momentum of car 1 is positive, and the momentum of car 2 is negative.
Now, we can apply the principle of conservation of momentum, which states that the total momentum before the collision is equal to the total momentum after the collision:
p1 + p2 = p_final
Considering that the collision is perfectly inelastic, the two cars stick together and move as one mass after the collision. Let p_final be the momentum of this new combined mass. The final momentum will have both magnitude and direction. To calculate it, we can transform it into a vector quantity. Let's define the final displacement of the combined mass as x_final (east), and y_final (south).
p_final = (m1 + m2) * v_final
We can express v_final as a vector quantity by breaking it down into its x-direction (v_x_final) and y-direction (v_y_final) components. The final displacement of the combined mass will be given by:
x_final = v_x_final * t
y_final = v_y_final * t
Here, t represents the time after the collision.
To find v_x_final and v_y_final, we need to use the concept of conservation of momentum.
Conservation of momentum in the x-direction:
p1x + p2x = p_finalx
Since car 2 only moves in the x-direction and car 1 only moves in the y-direction before the collision, the x-components of their momentum are:
p1x = 0 (no x-component)
p2x = -3500 kg * 20 m/s (negative because it is in the opposite direction)
So, we have:
-3500 kg * 20 m/s = (m1 + m2) * v_x_final
Conservation of momentum in the y-direction:
p1y + p2y = p_finaly
Since car 1 moves only in the y-direction and car 2 moves only in the x-direction before the collision, the y-components of their momentum are:
p1y = 1800 kg * 15 m/s (positive because it is in the same direction)
p2y = 0 (no y-component)
So, we have:
1800 kg * 15 m/s = (m1 + m2) * v_y_final
Now, we have two equations with two unknowns (v_x_final and v_y_final). We can solve for them simultaneously.
To find the final displacement of the cars after the collision, we will first calculate the total momentum before and after the collision. We can then use the principle of conservation of momentum to determine the final velocity and displacement.
Step 1: Calculate the initial momentum of each car.
The momentum of an object can be calculated using the formula: momentum = mass × velocity.
The initial momentum of the first car (car 1) traveling south is:
momentum1 = mass1 × velocity1
= 1800 kg × (-15 m/s) (negative sign indicates south direction)
= -27000 kg·m/s
The initial momentum of the second car (car 2) traveling east is:
momentum2 = mass2 × velocity2
= 3500 kg × 20 m/s
= 70000 kg·m/s
Step 2: Calculate the total initial momentum.
The total initial momentum is the sum of the momenta of the two cars before the collision:
total initial momentum = momentum1 + momentum2
= -27000 kg·m/s + 70000 kg·m/s
= 43000 kg·m/s
Step 3: Calculate the final velocity of the combined cars.
According to the principle of conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision.
Since the collision is perfectly inelastic, the two cars stick together and move as one object after the collision. Therefore, the final mass (m_f) of the combined cars will be the sum of the masses of the two cars.
m_f = mass1 + mass2
= 1800 kg + 3500 kg
= 5300 kg
Using the formula for momentum (momentum = mass × velocity), we can find the final velocity (v_f) of the combined cars.
total final momentum = (mass1 + mass2) × v_f
43000 kg·m/s = 5300 kg × v_f
v_f = 43000 kg·m/s / 5300 kg
v_f ≈ 8.1132 m/s (rounded to four decimal places)
Step 4: Calculate the coefficient of static friction force.
Since the cars are moving and the road is exerting a frictional force, we can calculate the force of friction using the formula: force of friction = coefficient of kinetic friction × normal force.
The normal force (N) can be calculated using the formula: normal force = mass × gravitational acceleration.
normal force = (mass1 + mass2) × gravitational acceleration
= 5300 kg × 9.8 m/s²
≈ 51940 N (rounded to the nearest whole number)
force of friction = coefficient of kinetic friction × normal force
= 0.8 × 51940 N
≈ 41552 N (rounded to the nearest whole number)
Step 5: Calculate the acceleration due to friction.
Using Newton's second law of motion (force = mass × acceleration), we can calculate the acceleration due to friction:
force of friction = (mass1 + mass2) × acceleration due to friction
41552 N = 5300 kg × acceleration due to friction
acceleration due to friction = 41552 N / 5300 kg
≈ 7.8274 m/s² (rounded to four decimal places)
Step 6: Calculate the time it takes for the cars to stop.
To find the time it takes for the cars to stop, we can use the equation of motion: final velocity (v_f) = initial velocity (v_i) + acceleration (a) × time (t).
We want to find the time it takes for the cars to stop, so the final velocity will be zero.
0 = 8.1132 m/s + (-7.8274 m/s²) × t (negative sign indicates deceleration)
7.8274 m/s² × t = 8.1132 m/s
t = 8.1132 m/s / 7.8274 m/s²
t ≈ 1.0375 s (rounded to four decimal places)
Step 7: Calculate the distance traveled during deceleration.
Using the equation of motion: distance (d) = initial velocity (v_i) × time (t) + 0.5 × acceleration (a) × time², we can calculate the distance traveled during deceleration.
distance = 8.1132 m/s × 1.0375 s + 0.5 × (-7.8274 m/s²) × (1.0375 s)²
≈ 4.2838 m (rounded to four decimal places)
Therefore, the final displacement of the cars after the collision, taking into account the deceleration caused by the friction between the tires and the road, is approximately 4.2838 meters to the south.