Calculating Time with Forces Acting on a Block

In summary, the conversation discusses the steps to find the minimum time required to travel a horizontal distance with a given acceleration. The solution involves maximizing the horizontal acceleration by focusing on values that maximize the function of sin(2θ). Alternatively, one can also maximize or minimize t by maximizing or minimizing t^2. The conversation also provides a tip for maximizing θ by letting u = tanθ and maximizing u.
  • #1
tryingtolearn1
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5
Homework Statement
A block starts at rest and slides down a frictionless plane inclined at an angle ##\theta##. What should ##\theta## be so that the block travels a given horizontal distance in the minimum amount of time?
Relevant Equations
$$F=ma$$
When drawing a diagram of the forces acting on the block, I have the following forces: $$\sum F_x = a_x = (g \sin\theta) \cos \theta .$$

Now, I can use the following kinematic equation $$x=vt+\frac{a_xt^2}{2}$$, where $$v=0$$ and $$a_x = (g \sin\theta) \cos \theta$$ $$\therefore \frac{2x}{t^2} = (g \sin\theta) \cos \theta .$$

Now in order to obtain the minimum amount of time, I will need to take the derivative and set it equal to zero but this is where I get stuck and don't know how to proceed?
 
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  • #2
What if you simply maximise the horizontal acceleration? Does that work?
 
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  • #3
Hi.

To travel a given horizontal distance (x) in the minimum amount of time (t) requires you to maximise horizontal acceleration. You have already found that [tex]a_x = (g \sin\theta) \cos \theta[/tex]Hint: [itex]sin(2A) = sinA cos(A) [/itex] and ask yourself for what value of A is sin(2A) maximum? No calculus needed!

Alternatively, if you are required to provide a calculus-based solution, express t as a function of [itex]\theta[/itex], treating x as a constant (since it is a 'fixed distance'). Then differentiate and solve [itex] \frac {dt}{d\theta} = 0[/itex] as usual.
 
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  • #4
Ah, I see thank you @PeroK and @Steve4Physics. The derivative expressing ##t## as a function of ##\theta## turned out to be messy where I had to solve the following: $$\frac{dt}{d\theta}=\sqrt{\frac{4x}{g\sin2\theta}}=0$$ so instead I just took the hint you provided of maximizing the horizontal acceleration by focusing on values that maximizes the function ##\sin2\theta## which is ##\frac{\pi}{4}##.
 
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  • #5
Here's another tip: maximising or minimising ##t## is the same as maximising or minimising ##t^2##. That can be very useful. Same for ##v## and ##v^2##.

The same if you have to maximise ##\theta## and you have a formula for ##\tan \theta##. Just let ##u = \tan \theta## and maximise ##u##.
 
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  • #6
tryingtolearn1 said:
Ah, I see thank you @PeroK and @Steve4Physics. The derivative expressing ##t## as a function of ##\theta## turned out to be messy where I had to solve the following: $$\frac{dt}{d\theta}=\sqrt{\frac{4x}{g\sin2\theta}}=0$$ so instead I just took the hint you provided of maximizing the horizontal acceleration by focusing on values that maximizes the function ##\sin2\theta## which is ##\frac{\pi}{4}##.
I assume you mean $$\frac{dt}{d\theta}=\frac{d}{d\theta}\sqrt{\frac{4x}{g\sin2\theta}}=0$$
 
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Related to Calculating Time with Forces Acting on a Block

What is the formula for calculating time with forces acting on a block?

The formula for calculating time with forces acting on a block is t = √(2m/F), where t is the time in seconds, m is the mass of the block in kilograms, and F is the net force acting on the block in Newtons.

How do you determine the mass of the block?

The mass of the block can be determined by using a scale or balance to measure its weight in kilograms. This can also be done by dividing the weight in Newtons by the acceleration due to gravity (9.8 m/s²).

What is the net force acting on the block?

The net force acting on the block is the sum of all the forces acting on it. This can be calculated by adding together all the individual forces, taking into account their direction and magnitude.

What is the unit of measurement for time in this calculation?

The unit of measurement for time in this calculation is seconds (s).

Can this formula be used for any type of force acting on a block?

Yes, this formula can be used for any type of force acting on a block as long as the mass of the block and the net force are known. It is important to make sure that the units for mass and force are consistent in the calculation.

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