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Here is the original question:

A car of mass $m$ is initially at rest on the boat of mass $M$ tied to the wall of a dock by a massless inextensible string. The car accelarates from rest to velocity $v$ in time $t$. The car then applies its brakes and comes to rest in negligible time. Neglect friction between the boat and water;the time at which the boat will strike the wall

I approached the question as follows:

from time=$0$ to time=$t$

I assumed that the force of tension (due to string), acts only on the centre of mass of car boat system. Then the accelaration of the car-boat system would be:

$$a_{car-boat}=\frac{m\left(a_\text{car with respect to water}\right)+M\left(a_{boat}\right)}{M+m}$$

Here is what I don't understand:

If the car and boat were not attached by string, then the boat would move backwards (with respect to reference frame attached to water). However, the boat moves forward due to the tension force in the string. The tension force only arises because of the car's motion, and causes accelaration of the centre of mass of the car-boat system.

Why does the boat move forward? If it moves forward, shouldn't the rope become slack and not exert any force on the boat?

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I'm first going to restate what I believe you meant to ask in your post... correct me if I misunderstood.

Scenario

Setup:

  • A car is at rest atop a boat
  • The boat is attached to a wall by a string

Situation:

  1. The car accelerates towards the wall (in direction of string)
  2. After some time $t$ the car stops quickly

Question:

Why does the boat move towards the wall?

Assumptions:

  • String cannot be stretched
  • No friction between boat and water

Understanding the problem

  1. The acceleration of the car

When the car is resting on the boat, we can find where their center of mass lies. If we then accelerate the car (assuming no losses), the boat is pushed in the opposite direction such that the center of mass stays in a fixed position. Note that the boat cannot be pushed away from the wall in this situation because it is tethered by a string.

  1. The stopping of the car

For the car to decelerate to rest (relative to the boat) it brakes, receiving a frictional force directed away from the wall and, as given by Newton's Third Law, exerting an equal force on the boat directed towards the wall.

  1. The acceleration of the boat

The boat will then be accelerated by the reaction force exerted on it, caused by the car's braking, causing it to begin moving towards the wall.

  1. The motion of the boat

The boat is then simply traveling with uniform velocity towards the wall, with which it will collide after some amount of time.


Calculation process

  1. The acceleration of the car

The actual magnitude of the acceleration is not important for finding the final velocity of the boat, so all we are concerned with is the velocity of the car after the acceleration, which we already know to be:

$$v_{\text{car}}=v$$

2/3. The stopping of the car/acceleration of the boat

Conservation of momentum allows us to calculate the velocity of the boat this problem simply enough:

$$p_f=p_i$$ $$(m_{\text{boat}}+m_{\text{car}})v_{\text{boat}_f}=m_{\text{car}}v_{\text{car}_i}$$ $$v_{\text{boat}_f}=\frac{m_{\text{car}}v_{\text{car}_i}}{m_{\text{boat}}+m_{\text{car}}}$$

  1. The motion of the boat

This is now just a problem of finding the time given a uniform velocity:

$$t=\frac{d}{v_{\text{boat}}}$$

where $d$ is the distance to the wall.

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  • $\begingroup$ Shouldn't the final momentum also consider the car's momentum, since it is moving with the boat? $\endgroup$ – Níckolas Alves Nov 5 '17 at 6:07
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    $\begingroup$ @NickolasAlves Thanks for pointing that out... Yes, it should. $\endgroup$ – Kieran Moynihan Nov 5 '17 at 6:08
  • $\begingroup$ Thank you so much for this answer! One clarification though: If the forward motion is caused by the braking and not tension developed, then does it move at all before braking? Also to conserve momentum, shouldn't the net external force be zero(i.e doesn't the tension in string exert a force on the system)..Thanks again! $\endgroup$ – GreenApple Nov 5 '17 at 6:17
  • $\begingroup$ I'll answer your questions in reverse order. In reality, the buildup of tension in the string would cause some stretching, and a spring effect; however, since the string cannot be stretched, it acts as a wall, and cannot do anything other than stop the boat from moving away from its tether. In answer to your first question: no, because, as I said for your second question, the string prevents the boat from being accelerated by the car. $\endgroup$ – Kieran Moynihan Nov 5 '17 at 6:25
  • $\begingroup$ Additionally, note that the center of mass will, rather than remaining fixed—as it would if the boat was free of its tether—shift as the car moves along the boat. $\endgroup$ – Kieran Moynihan Nov 5 '17 at 6:27

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