whow long will it take for the bee to smash into your windshield

Question

when travelling on your motorcycle at a velocity of 70.0km/h (E), you notice a bumble bee 10.0m in front of you is flying 2.90m/s (W) If both you and the bee are travelling in a straight path, how long will it take for the bee to smash into your windshield?

The answer is 2.24s and I think you are supposed to get the answer by doing t=50/19.44+2.90 I want to know why the two velocities are added together. Shouldn't they would be subtracted because of the opposite directions?

Please show work and explain thanks!

Answer 1

Hi!

First of all, good on you for asking for help on how to understand the problem instead of just asking for the answer! It'll help a lot more in the long run! Now, onto the problem:

When you're having trouble dealing with positives and negatives, it may help to define a coordinate system. In this case, let's say the we consider East to be the positive direction and West to be the negative direction. This would mean that the velocity of you on your motorcycle (v_M) is +70 km/h and the velocity of the bumblebee (v_B) is -2.90 m/s. First, we should convert the motorcycle velocity to units of m/s:

v_M = +70.0 km / hr * 1000 m / km * (1 hr / 60 min) * (1 min / 60 s) = +19.444 m / s (note that this number should be reported with three significant figures, but let's hang on to a couple more until our calculations are done to avoid rounding errors)

To find the relative velocity between two objects, you take the difference between the two velocities. We can do this in either order, but there is a slight difference in what the result means:

v_relative = v_M - v_B = 19.444 m/s - (-2.90 m/s) = +22.344 m/s

v_relative = v_B - v_M = (-2.90 m/s) - 19.444 m/s = -22.344 m/s

(this result should also be considered to have 3 significant figures, but again let's save a couple more for our calculations)

So what does each one mean? When calculating the relative velocity of two object, the first term is the object we consider to have the velocity, and the second term is the object we consider to be the point of reference. So, our first result is the velocity of the motorcycle from the bumblebee's point of view, and the second result is the velocity of the bumblebee from the motorcycle's point of view.

We could use the same calculation for any other object as well. Suppose there's a stationary cactus on the side of the road. The cactus would have a velocity (v_C) of 0 m/s, from a stationary viewpoint. Let's say we want to know the relative velocity of the cactus from the bumblebee's point of view; the calculation would be:

v_relative = v_C - v_B = 0 m/s - (-2.90 m/s) = +2.90 m/s

So, from the bumblebee's perspective, the cactus appears to be traveling 2.90 m/s in the East direction.

Anyways, getting back to the problem, we know that the bumblebee and motorcycle are 10.0 m apart and are approaching each other at 22.344 m/s. We can calculate the time it takes for them to collide with Distance = Rate * Time. Solving for time, we get Time = Distance / Rate = 10.0 m / 22.344 m/s = 0.4475 s.

This doesn't line up with your given answer, but I think it may be the result of some confusion with the givens. You provide the calculation t = 50 / (19.4 + 2.90), which seems to indicate a distance of 50 meters instead of 10 meters. 50 meters divided by 22.344 meters per second is 2.24 seconds, so if the bumblebee and motorcycle are really 50.0 m apart, then:

Time = 50.0 m / 22.344 m/s = 2.24 s

Let me know if you're still having trouble / have any questions!

Brief Side Note: The relativity calculation discussed above only works for classical or Newtonian mechanics. If you start dealing with special relativity, it gets harder and weirder!