Two vehicles collide and stick together. After the collision, their combined y-momentum is 2.40 × 104 kilogram meters/second, and their x-momentum is 7.00 × 104 kilogram meters/second. What is the angle of the motion of the two vehicles, with respect to the x-axis?
arctangent (Vy/Vx) = 18.9 degrees
To find the angle of motion of the two vehicles with respect to the x-axis, we can use the concept of vector addition.
1. Start by breaking down the combined momentum vector into its x and y components.
- Given: Combined y-momentum = 2.40 × 10^4 kilogram meters/second
Combined x-momentum = 7.00 × 10^4 kilogram meters/second
The y-component of the momentum is 2.40 × 10^4 kilogram meters/second, and the x-component is 7.00 × 10^4 kilogram meters/second.
2. Use the components of the momentum vector to find the magnitude of the combined momentum vector using the Pythagorean theorem:
- Magnitude of the combined momentum vector (P) = √(Px^2 + Py^2)
P = √((7.00 × 10^4)^2 + (2.40 × 10^4)^2)
3. Solve the equation to find the magnitude:
P = √(49 × 10^8 + 5.76 × 10^8)
= √(54.76 × 10^8)
≈ √(55 × 10^8)
= 2.34 × 10^4 kilogram meters/second
4. Now, to find the angle of motion (θ), we can use the arc tangent function:
- θ = tan^(-1)(Py/Px)
θ = tan^(-1)((2.40 × 10^4)/(7.00 × 10^4))
5. Calculate the angle θ:
θ = tan^(-1)(0.3429)
≈ 19.24 degrees
Therefore, the angle of motion of the two vehicles, with respect to the x-axis, is approximately 19.24 degrees.
To find the angle of motion of the two vehicles with respect to the x-axis, we need to use the concept of vector addition and trigonometry.
Let's denote the y-component of the momentum as Py and the x-component of the momentum as Px. According to the problem, Py (combined y-momentum) is 2.40 × 10^4 kilogram meters/second, and Px (combined x-momentum) is 7.00 × 10^4 kilogram meters/second.
We can use these momentum components to find the resultant momentum vector using the Pythagorean theorem. The magnitude of the resultant momentum vector is given by:
Mag(R) = sqrt(Px^2 + Py^2)
Substituting the given values:
Mag(R) = sqrt((7.00 × 10^4)^2 + (2.40 × 10^4)^2)
Now, we can use the trigonometric functions to find the angle of the motion. The angle θ is given by:
θ = arctan(Py / Px)
Substituting the values of Px and Py:
θ = arctan(2.40 × 10^4 / 7.00 × 10^4)
Now we can evaluate this expression to find the angle θ using a scientific calculator or an online calculator.
Note: Make sure to convert the angle from radians to degrees if required.
By following these steps, you should be able to determine the angle of motion of the two vehicles with respect to the x-axis.