Question

a train is traveling east at a constant speed (v_t), as shown in the diagram. a cyclist moving at constant speed (v_c) heading 15° east of north with respect to the ground is also shown. if (v_c < v_t), which of the following vectors could represent the velocity of the cyclist as measured from an observer in car 1 and from an observer in car 3? a (v_c) as measured from car 1 (v_c) as measured from car 3 b (v_c) as measured from car 1 (v_c) as measured from car 3 c (v_c) as measured from car 1 (v_c) as measured from car 3 d (v_c) as measured from car 1 (v_c) as measured from car 3

Explanation:

Step1: Understand relative - velocity concept

The velocity of the cyclist with respect to an observer in the train is given by $\vec{v}_{C - T}=\vec{v}_C-\vec{v}_T$. The train is moving east - ward with speed $v_T$ and the cyclist is moving $15^{\circ}$ east of north with speed $v_C$ ($v_C < v_T$).

Step2: Analyze observer in Car 1

For an observer in Car 1 (assuming Car 1 is part of the train), the east - ward component of the cyclist's velocity relative to the train will be $v_{Cx}-v_T$ (where $v_{Cx}=v_C\sin15^{\circ}$) and the north - ward component is $v_{Cy} = v_C\cos15^{\circ}$. Since $v_T>v_C$, the east - ward component of the relative velocity will be negative. So the relative velocity vector will have a west - ward and north - ward component.

Step3: Analyze observer in Car 3

For an observer in Car 3 (assuming Car 3 is part of the train), the relative velocity of the cyclist has the same vector characteristics as for the observer in Car 1 because the relative motion of the cyclist with respect to the train is the same for all points on the train.

Answer:

C. $v_C$ as measured from Car 1 (vector with west - ward and north - ward components) and $v_C$ as measured from Car 3 (vector with west - ward and north - ward components)