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I was thinking about the following, a block on an inclined plane no friction. However this inclined plane is in a truck accelerating forward. Note: the inclined plane doesn't slip. So I wondered what the following would be

a) the acceleration of the block with respect to someone inside the truck and

b) the acceleration of the block with respect to someone standing on the ground beside the truck

c) Do they both agree on the time it takes to hit the floor of the truck?

d) what horizontal distance does the block cover in both of the reference frames?

My thoughts: After drawing free body diagrams and calculating I'm pretty sure accelerations are the same. For now I think the times will be the same. That being said what happens to the distance covered, do I have to include the distance covered by the truck?

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Answer to a- the only force acting (as stated friction is not present)is weight of it and the body will have only acceleration due to gravity more accurately saying only have vertical component of acceleration due to gravity.

Answer to b - acceleration as seen by some one on ground will be vector sum of pseudo acceleration due to acceleration of the truck and acceleration due to gravity.

Answer to c - as both acceleration are different so the time calculated but it also depends upon angle of inclination.

Answer to d - distance will not be same in both cases it will be different as distance traveled by truck will be added.

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I think of Newton's First Law as a definition of an inertial frame (an inertial frame is one in which an object with no net force moves with a constant velocity) and a statement that one must be in an inertial frame to apply Newton's Laws.

Therefore, for question (a), the person inside the truck should first test whether they are in an inertial frame: place a ball on the ground; it evidently has zero net force applied to it, but it will be seen to accelerate toward the back of the truck. This "ghost force" indicates that the person is not in an intertial frame and should therefore not attempt to use Newton's Second Law.

The person standing outside the truck is in an inertial frame (they too can do an "inertial frame test" and find, to within the precision of their measurements, the ground on Earth is inertial), so they are free to apply Newton's Second Law. They must however choose a coordinate system rooted to their frame, so the acceleration of the block will be expressed as the second time derivative of the vector sum of the positions $\vec{R}$ and $\vec{r}$, where $\vec{R}$ is the (horizontal) position vector of the truck with respect to the ground and $\vec{r}$ is the usual position of the block relative to the top of the incline.

This question considers the special case that the block does not move with respect to the incline. Whether the block slides down the incline, up the incline, or does not slide at all, depends on the specifics of the problem (the acceleration of the truck and the angle of the incline).

So, for questions (b) and (d), the answer is no they will not agree. The observer on the ground will say that the acceleration of the block is approximately the same as the truck while the observer in the truck might see any kind of accleration along the incline. The horizontal distance covered from the perspective of the observer on the ground is again very similar to that covered by the truck, while for the observer in the truck it could be many things.

For question (c), the block might never reach the bottom of the incline (or might fall off the top!). But any timing of events will be the same for the two observers (in a non-relativistic situation).

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Analysis

Fist question

The truck's frame is non-inertial. So while calculating, you have to take in account the pseudo-force acting in the opposite horizontal direction.

Second question

You can use the result of the first case to calculate the relative accelrations in second case.

Third question

Time taken is irrespective of the observer (atleast in this case).

Fourth question

Horizontal displacent in truck's frame would be added to the displacement of the truck (in ground frame) to get the dispolacement of block in ground frame.

PS: I've given hints to solve the questions. If you still cannot solve it, let me know.

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a) the acceleration of the block with respect to someone inside the truck

Lets label the angle between the slope and the horizontal as $\theta$. The acceleration of the block down the slope due to gravity (g) is $ g\sin \theta$. The horizontal component of that acceleration in the forwards direction is $g \cos\theta \sin \theta $. The acceleration backwards due to the inertial force of the block is the negative of the acceleration of the truck (a) so the total horizontal acceleration is $g \cos\theta \sin \theta - a $ inside the truck.

b) the acceleration of the block with respect to someone standing on the ground beside the truck

To this observer, the forwards acceleration due to gravity is the same as above. The inertial acceleration of the block vanishes and there is only the horizontal acceleration of the truck acting in the forwards direction, so the net acceleration in the ground frame is $g \cos\theta \sin \theta + a $

c) Do they both agree on the time it takes to hit the floor of the truck?

Yes, since the question is in the context of Newtonian physics, the time between two events is always the same in any reference frame, inertial or not. It is worth noting that if $g \cos\theta \sin \theta < a $ then the block will slide upwards instead of downwards. If the block is sliding downwards and the length of the slope is x, then the time for the block to slide that distance down the slope is $t = \sqrt{\frac{2x}{g\sin \theta}} $. (This is obtained from the equation of motion given below.)

d) what horizontal distance does the block cover in both of the reference frames?

Using the equation of motion $d = 1/2 a t^2$, the relevant distances are found by plugging in the respective horizontal accelerations given above.

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A) With respect to someone inside the truck, the block will be accelerating(appear to accelerate) in opposite direction to that of truck.So its acceleration wrt to someone in truck will be equal in magnitude but in opposite direction of truck's accleration.

B)WRT someone on the ground the block has no horizontal acceleration.

C)Time to fall down is same as you think.

D)

For someone outside,

Horizontal acceleration = 0 (as stated in B)

Therefore, horizontal distance =0

Similarly, for someone inside,

Horizontal acceleration = negative (as stated in A)

Therefore, horizontal distance =negative

whose magnitude can be calculated if acceleration of truck and time taken to fall on ground in known.

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    $\begingroup$ How can the block have no horizontal acceleration with respect to the ground. If it's falling down the inclined plane there is an acceleration component acting horizontally. $\endgroup$
    – Sigmund Reed
    Commented Apr 2, 2017 at 15:08
  • $\begingroup$ The answers to A and B are wrong. Take the special case where the acceleration of the truck is 0. The block still experiences a normal force from the inclined plane with a horizontal component. $\endgroup$
    – Brick
    Commented Jul 1, 2019 at 14:26

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