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Why does a ball inside a moving bus at rest start moving backwards when the bus suddenly accelerates? Also does the moving ball have some acceleration?

This is my theory: Initially the bus and the ball are at rest. When the bus starts accelerating, due to inertia of rest, the ball resists change in motion and tends to remain at rest. Since it is an accelerating frame, pseudo forces of unknown origin act on the ball in the direction opposite to the direction of motion of the bus. This pseudo force is responsible for the ball to move in the opposite direction with some acceleration. This explanation with respect to the accelerating frame.

But how do I explain this fact from the frame of reference of a person on the road? I understand that the acceleration of the ball is in the same direction as that of acceleration of bus. There is no other force to balance this force which is in the direction of motion of bus. So the ball should have moved in a direction along the direction of the bus. But the reverse happens. Please help me. enter image description here

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    $\begingroup$ From the non inertial frame its the pseudo force. From the inertial its inertia $\endgroup$ – Shashaank Mar 4 '17 at 14:51
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    $\begingroup$ If you neglect friction ball wont move unless it hits the back of the bus and gets dragged $\endgroup$ – SaudiBombsYemen Mar 4 '17 at 15:50
  • $\begingroup$ You answered your question. There is nothing left to answer. Your theory is correct. $\endgroup$ – Yashas Mar 4 '17 at 16:19
  • $\begingroup$ By the way Pseudo forces are as real as imaginary numbers are real !! $\endgroup$ – Shashaank Mar 4 '17 at 18:46
  • $\begingroup$ What makes you say that Shashank? $\endgroup$ – Aditya DS Mar 4 '17 at 19:04
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Analyse it this way .

If the bus is infinitely long such that there is no force from the walls and there is no friction also. Then in the road frame the ball stays where it is ( because no force acts on it ) while the bus moves forward.

In the bus frame the bus is at rest while to the bus the ball accelerates back which you have explained by the pseudo force .

Friction enters then rolling enters the scene sort of complicating the scene .

If there is Friction

Let u be the coefficient of friction then

From ground frame the FBD of the ball is

f=uN=ma towards the right.

From accelerated bus frame

f=uN=m(a+A) (towards the left) mA being the pseudo force

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  • $\begingroup$ Why don't you take friction into account and explain it? Yours is an ideal case. I want an explanation that fits in the real world $\endgroup$ – Aditya DS Mar 5 '17 at 5:28
  • $\begingroup$ @AdityaDS because what Sammy gerbil said for the real case is right $\endgroup$ – Shashaank Mar 5 '17 at 5:31
  • $\begingroup$ Didn't you read my next two comments? $\endgroup$ – Aditya DS Mar 5 '17 at 5:31
  • $\begingroup$ @AdityaDS There is nothing to explain . What he said is precise and right and it's already explained I guess. How can I force to imagine that. You need to analyse the situation more yourself understand it . If friction acts on the ball has to accelerate with the bus . $\endgroup$ – Shashaank Mar 5 '17 at 5:34
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Take it like this imagine yourself as an by stander watching the bus then you would notice the ball goes in the direction opposite to bus but when we only take the ball the direction of pseudo force is opposite to the motion of the bus when we consider the floor to have friction then this $μN$ force to opposethis pesudo force hence the real motion occurs in the opposite direction of bus.while considering the consequences the motion of ball by the by stander is same as the one in the bus watching the ball.

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