Question

solution
find the x - components of the initial and final total momenta.
set the initial x - momentum equal to the final x - momentum.
find the y - components of the initial and final total momenta.
set the initial y - momentum equal to the final y - momentum.
divide equation (2) by equation (1) and solve for θ.
substitute this angle back into equation (2) to find ( v_f ).
learn more
remarks its also possible to first find the x - and y - components ( v_{fx} ) and ( v_{fy} ) of the resultant velocity. the magnitude and direction of the resultant velocity can then be found with the pythagorean theorem, ( v_f=sqrt{v_{fx}^2 + v_{fy}^2} ), and the inverse tangent function ( \theta=\tan^{-1}(v_{fy}/v_{fx}) ). setting up this alternate approach is a simple matter of substituting ( v_{fx}=v_fcos\theta ) and ( v_{fy}=v_fsin\theta ) in equations (1) and (2).
question if the car and van had identical mass and speed, what would the resultant angle have been?

Explanation:

Step1: Calculate initial x - momentum

$$\sum p_{xi}=m_{car}v_{car}$$
Given \(m_{car} = m_{van}=m\) (since they have identical mass), \(v_{car}=v_{van}=v\) (identical speed).
Initial \(x -\)momentum \(\sum p_{xi}=mv\)

Step2: Calculate final x - momentum

$$\sum p_{xf}=(m + m)v_f\cos\theta=2mv_f\cos\theta$$
Set \(\sum p_{xi}=\sum p_{xf}\), so \(mv = 2mv_f\cos\theta\)

Step3: Calculate initial y - momentum

$$\sum p_{yi}=m_{van}v_{van}=mv$$

Step4: Calculate final y - momentum

$$\sum p_{yf}=(m + m)v_f\sin\theta=2mv_f\sin\theta$$
Set \(\sum p_{yi}=\sum p_{yf}\), so \(mv=2mv_f\sin\theta\)

Step5: Divide y - momentum equation by x - momentum equation

$$\frac{\sum p_{yf}}{\sum p_{xf}}=\frac{mv}{mv}=\frac{2mv_f\sin\theta}{2mv_f\cos\theta}$$
$$1 = \tan\theta$$

Answer:

\(\theta = 45^{\circ}\)