A 1100- kg car collides with a 1500- kg car that was initially at rest at the origin of an x-y coordinate system. After the collision, the lighter car moves at 20.0 km/h in a direction of 35 o with respect to the positive x axis. The heavier car moves at 23 km/h at -50 o with respect to the positive x axis.
What was the initial speed of the lighter car (in km/h)?
What was the initial direction (as measured counterclockwise from the x-axis)?
To find the initial speed of the lighter car, we need to use the law of conservation of momentum. According to this law, the total momentum before the collision is equal to the total momentum after the collision.
The momentum of an object can be calculated by multiplying its mass by its velocity.
Before the collision:
Momentum of the lighter car (m1) = mass of the lighter car (m1) * initial velocity of the lighter car (v1)
Momentum of the heavier car (m2) = mass of the heavier car (m2) * initial velocity of the heavier car (v2)
After the collision:
Momentum of the lighter car (m1') = mass of the lighter car (m1) * final velocity of the lighter car (v1')
Momentum of the heavier car (m2') = mass of the heavier car (m2) * final velocity of the heavier car (v2')
According to the problem, the heavier car was initially at rest. Therefore, the initial velocity of the heavier car (v2) is 0.
Using the law of conservation of momentum, we can write the equation:
(m1 * v1) + (m2 * v2) = (m1' * v1') + (m2' * v2')
Substituting the given values:
(1100 kg * v1) + (1500 kg * 0) = (1100 kg * 20.0 km/h * cos(35°)) + (1500 kg * 23 km/h * cos(-50°))
Simplifying the equation:
(1100 kg * v1) = (1100 kg * 20.0 km/h * cos(35°)) + (1500 kg * 23 km/h * cos(-50°))
Dividing both sides by 1100 kg:
v1 = (20.0 km/h * cos(35°)) + (1500 kg * 23 km/h * cos(-50°))
Calculating the value:
v1 ≈ 20.0 km/h * cos(35°) + 1500 kg * 23 km/h * cos(-50°)
Using a calculator, we find that:
v1 ≈ 11.32 km/h + 16.01 km/h ≈ 27.33 km/h
Therefore, the initial speed of the lighter car was approximately 27.33 km/h.
Now let's find the initial direction (as measured counterclockwise from the x-axis). The initial direction can be found by using the inverse tangent function of the ratio of the y-component to the x-component of the lighter car's initial velocity.
The y-component can be calculated by multiplying the initial speed (v1) by the sine of the angle (35°).
y-component = v1 * sin(35°)
y-component = 27.33 km/h * sin(35°)
Using a calculator, we find that the y-component is approximately:
y-component ≈ 27.33 km/h * sin(35°) ≈ 15.55 km/h
The x-component can be calculated by multiplying the initial speed (v1) by the cosine of the angle (35°).
x-component = v1 * cos(35°)
x-component = 27.33 km/h * cos(35°)
Using a calculator, we find that the x-component is approximately:
x-component ≈ 27.33 km/h * cos(35°) ≈ 22.32 km/h
Now, we can find the direction by using the inverse tangent function:
Direction (θ) = atan(y-component / x-component)
Direction (θ) = atan(15.55 km/h / 22.32 km/h)
Using a calculator, we find that the direction (measured counterclockwise from the x-axis) is approximately:
Direction (θ) ≈ atan(15.55 km/h / 22.32 km/h) ≈ 36.13°
Therefore, the initial direction (as measured counterclockwise from the x-axis) is approximately 36.13°.
To find the initial speed and direction of the lighter car, we can use conservation of momentum.
1. Momentum is a vector quantity defined as the product of an object's mass and its velocity: p = m * v. Momentum is conserved in a collision, which means that the total momentum before and after the collision is the same.
2. Let's denote the initial velocity of the lighter car as v1 and the initial velocity of the heavier car as v2.
3. The total momentum before the collision is given by: p_initial = m1 * v1 + m2 * v2, where m1 and m2 are the masses of the lighter and heavier cars, respectively.
4. The total momentum after the collision is given by: p_final = m1 * v1_final + m2 * v2_final, where v1_final and v2_final are the final velocities of the lighter and heavier cars, respectively.
5. According to the problem statement, the heavier car was initially at rest (v2 = 0 km/h) and the final velocities of both cars are known.
6. Now we can set up the conservation of momentum equation: p_initial = p_final.
m1 * v1 + m2 * v2 = m1 * v1_final + m2 * v2_final
1100 kg * v1 + 1500 kg * 0 km/h = 1100 kg * 20.0 km/h * cos(35°) + 1500 kg * 23.0 km/h * cos(-50°)
7. Simplify and solve the equation to find v1.
1100 kg * v1 = 1100 kg * 20.0 km/h * cos(35°) + 1500 kg * 23.0 km/h * cos(-50°)
v1 = (1100 kg * 20.0 km/h * cos(35°) + 1500 kg * 23.0 km/h * cos(-50°)) / 1100 kg
8. Calculate v1 to find the initial speed of the lighter car.
v1 ≈ 22.55 km/h
9. To find the initial direction of the lighter car, we need to determine the direction as measured counterclockwise from the x-axis.
According to the problem statement, the lighter car moves at 20.0 km/h in a direction of 35° with respect to the positive x-axis.
Therefore, the initial direction of the lighter car is 35° counterclockwise from the x-axis.