physics acceleration components?

Question

The positions of four cars on four different roads are indicated by the dots. Which cars could possibly have the acceleration shown?

The answers are A, B, and C. I chose D, so...yeah I'm pretty lost.

Answer 1

The centripetal acceleration is directed towards the centre of a circle.  The linear acceleration is directed in the direction of motion.

In D the centripetal acceleration is to the right and the linear acceleration is down. So the net must be down and to the right.

In C the direction of motion is in the line left to right.  As long as the acceleration is in the left direction that is a possibility.

In A the circular motion alone would direct it in the direction to the left.

In B the circular motion would cause the acceleration to be UP and to the left but if we ALSO add acceleration along the path ( down and to the left ) we CAN get values such that the net is also to the left.  So A, B and C are POSSIBLE ( depending on values) but D is directed in the wrong direction.

Answer 2

The point is not to guess at random. If you spend 2 seconds looking at the question, you know C has to be one of the answers.





Acceleration is the change in velocity divided with the change in time.

First play with vector subtraction.Draw a possible velocity vector for a car on a certain point on its trajectory. Then imagine the car continuing in either direction and draw the second acceleration vector. The difference between these two vectors is where your average acceleration vector points.After getting a feel for this, consider decomposition of vectors into two components (like x and y) whose sum gives the original vector. For kinematics it is often useful to decompose acceleration into a component parallel to the current direction of motion and the one perpendicular to it.



We call the parallel component "tangential acceleration" and the perpendicular "centripetal acceleration".





Consider car A at the given instant going down.

You will notice that the centripetal acceleration must point left, because now there is no horizontal velocity but after a little while the velocity will have a horizontal component to the left.



The centripetal acceleration can't point to the right because that would mean the velocity is increasing to the right.

The tangential acceleration will point down (in the same direction as velocity) if A is speeding up, and up if slowing down.

So combining the centripetal and tangential acceleration into the whole acceleration vector, we see it has a 180° range of motion. It can point diagonally down and to the left, up and to the left, or directly to the left.

Answer 3

A (toward the center of the arc) and C (to the left) could both have the acceleration indicated by the orange arrow.



D's acceleration is to the right, and B's is toward the northwest.

Answer 4

I can't easily see any way D could have that acceleration and stay on the road. All of the others could.



Remember that acceleration is how velocity is changing.

Velocity is speed and direction. If a car stays on a road, direction has to be the direction of the road. But as the car moves around a curve, the direction of the road changes. Speed can also change. Both of those changes are acceleration.



Car C is the easiest. Its velocity can only be to the right or left. As C's speed changes, its velocity becomes more to the left or right. Therefore acceleration is to the left or right (or is zero), no up/down component.



Car A currently has velocity in either the up or down direction. But no matter which way it's going around the curve, its direction is becoming more to the left. Trace the curve either way, pointing your finger in the direction the car is facing at each moment: you go from pointing somewhat right to pointing somewhat left.

If car A has constant speed, then its acceleration is directly to the left. If A does not have constant speed, then its acceleration has a vertical component.



Likewise car D has acceleration to the right, possibly combined with a vertical component.



And car B has acceleration towards the inside of its curve, possibly combined with a perpendicular component. If B is speeding up while traveling clockwise or slowing down while traveling counterclockwise, its resultant acceleration could be to the left.