Let's take a few minutes to review the car problems using our newly defined equations: \(\small\mathsf{ \overrightarrow{v_{AB}} = \overrightarrow{v_{AC}} - \overrightarrow{v_{BC}} }\) or \(\small\mathsf{ \overrightarrow{v_{BA}} = \overrightarrow{v_{BC}} - \overrightarrow{v_{AC}} }\). In problems like these, it is really important to define what each velocity given in the problem is, and how to arrange the equation so that it will give you the correct answer. In other words, what's A, what's B, what's C, and what are \(\small\mathsf{ \overrightarrow{v_{AB}} }\), \(\small\mathsf{ \overrightarrow{v_{AC}} }\), and \(\small\mathsf{ \overrightarrow{v_{BC}} }\). That way, you know you can simply plug the values into the equation to know what mathematics you need to do. We will concentrate on one-dimensional problems to eliminate the complexity of two-dimensional vectors (for now). All of the following problems deal with cars; however, it could just as easily be different objects with velocities in the same dimension.
| Problem | Picture | Given/Find | Equation | Solution |
|---|---|---|---|---|
| You are sitting in a car that is moving at 25.4 m/s with respect to the road. Another car zooms past your car going the same direction at a speed of 31.6 m/s with respect to the road. What is the relative velocity of the zooming car with respect to your car? |
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\(\small\mathsf{ \overrightarrow{v_{BA}} = ? }\) \(\small\mathsf{ \overrightarrow{v_{AC}} = 24.5 \text{ m/s} }\) \(\small\mathsf{ \overrightarrow{v_{BC}} = 31.6 \text{ m/s} }\) |
\(\small\mathsf{ \overrightarrow{v_{BA}} = \overrightarrow{v_{BC}} - \overrightarrow{v_{AC}} }\) | \(\small\mathsf{ \overrightarrow{v_{BA}} = 31.6 \text{ m/s} - 25.4 \text{ m/s} }\) \(\small\mathsf{ \overrightarrow{v_{BA}} = 6.2 \text{ m/s} }\) The velocity of car B with respect to A is 6.2 m/s. |
| You are sitting in a car that is moving at 25.4 m/s with respect to the road. Another car zooms toward you at a speed of 31.6 m/s with respect to the road. What is the relative velocity of the zoom car with respect to your car? |
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\(\small\mathsf{ \overrightarrow{v_{BA}} = ? }\) \(\small\mathsf{ \overrightarrow{v_{AC}} = 24.5 \text{ m/s} }\) \(\small\mathsf{ \overrightarrow{v_{BC}} = -31.6 \text{ m/s} }\) |
\(\small\mathsf{ \overrightarrow{v_{BA}} = \overrightarrow{v_{BC}} - \overrightarrow{v_{AC}} }\) | \(\small\mathsf{ \overrightarrow{v_{BA}} = -31.6 \text{ m/s} - 25.4 \text{ m/s} }\) \(\small\mathsf{ \overrightarrow{v_{BA}} = -57.0 \text{ m/s} }\) The velocity of car B with respect to A is -57.0 m/s. |
| Your car's speedometer says that you are travelling at 21. 5 m/s. Another car zooms toward you with a relative velocity of 52.3 m/s. What is the speed of the zooming car with respect to the road? |
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\(\small\mathsf{ \overrightarrow{v_{BA}} = -52.3 \text{ m/s} }\) \(\small\mathsf{ \overrightarrow{v_{AC}} = 21.5 \text{ m/s} }\) \(\small\mathsf{ \overrightarrow{v_{BC}} = ? }\) |
\(\small\mathsf{ \overrightarrow{v_{BA}} = \overrightarrow{v_{BC}} - \overrightarrow{v_{AC}} }\) | \(\small\mathsf{ -52.3 = \overrightarrow{v_{BC}} - 21.5 \text{ m/s} }\) \(\small\mathsf{ -52.3 \text{ m/s} + 21.5 \text{ m/s} = \overrightarrow{v_{BC}} }\) \(\small\mathsf{ \overrightarrow{v_{BC}} = -30.8 \text{ m/s} }\) The velocity of car B with respect to the road is -30.8 m/s. |
| Your car's speedometer says that you are travelling at 21.5 m/s. You approach another car that is moving in the same direction as you. It appears that the car is moving at a relative speed of -4.2 m/s. What is the velocity of that car with respect to the road? |
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\(\small\mathsf{ \overrightarrow{v_{BA}} = -4.2 \text{ m/s} }\) \(\small\mathsf{ \overrightarrow{v_{AC}} = 21.5 \text{ m/s} }\) \(\small\mathsf{ \overrightarrow{v_{BC}} = ? }\) |
\(\small\mathsf{ \overrightarrow{v_{BA}} = \overrightarrow{v_{BC}} - \overrightarrow{v_{AC}} }\) | \(\small\mathsf{ -4.2 = \overrightarrow{v_{BC}} - 21.5 \text{ m/s} }\) \(\small\mathsf{ -4.2 \text{ m/s} + 21.5 \text{ m/s} = \overrightarrow{v_{BC}} }\) \(\small\mathsf{ \overrightarrow{v_{BC}} = 17.3 \text{ m/s} }\) The velocity of car B with respect to the road is 17.3 m/s. |
