Solve Kinematics Problem Homework: Part B

In summary: The attempt at a solution says that part b is having trouble with and they have a solution for part a. They are substituting a into the equation to get the final answer of v=a+at. They think that the integral of adt is equal to v, but they are not sure because they are not a regular user of Mathematica.
  • #1
Precursor
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Homework Statement
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The attempt at a solution

I have already solved for part a. It's part b that I'm having trouble with right now.

Here is what I have for part b:

[tex]\frac{\mathrm{d} v}{\mathrm{d} t} = a_{0} + kv[/tex]

[tex]dv = a_{0}dt + kvdt[/tex]

[tex]dv = a_{0}dt + adt[/tex]

[tex]\int dv = \int a_{0}dt + adt[/tex]

And my final answer is:

[tex]v = a_{0}t + at[/tex]

Is this correct?
 
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  • #2
I don't understand the 3rd step where you replace kv with a.
Are you assuming that the acceleration is kv ?

Also in the final step, you assume that "a" is a constant.
Acceleration can't be constant; it is proportional to the velocity and the velocity is decreasing.

I have forgotten how to solve differential equations, but you might try a substitution to make it look easier. If you let v = u + b where b is a constant, then your dv/dt = ao + kv will change to
du/dt = ao +kb + ku
and you can choose b = -ao/k so
du/dt = ku.
This is the equation for radioactive decay, and it can easily be integrated to get u as an exponential function. Change variables back to v to get an exponential function for v.
 
  • #3
I made kv = a because the dimension of k is 1/T, where T is time, and v is simply velocity. Therefore, the product of these two variables will be acceleration.
 
  • #4
But kv is not a constant, so your "a" is not a constant and its integral over time is not at.
 
  • #5
Does this mean that the integral of adt is equal to v.
 
  • #6
Your "a" is really "k*v" so its integral over time would be k*distance + constant, where distance is a function of time. I don't think this does any good. Better to leave the kv in the differential equation and make the substitution I suggested in the 2nd post.
 
  • #7
I don't think your integration is correct. Go to Wolframalpha and use Mathematica's DSolve command for v'[t]=a+k*v[t] I think the solution is v[t]=-a/k + e^(k*t) C1 but not 100% sure since I'm not a regular user of Mathematica
 

FAQ: Solve Kinematics Problem Homework: Part B

How do I approach solving a kinematics problem?

Solving a kinematics problem involves using the equations of motion and setting up a problem-solving strategy. This includes identifying the known and unknown variables, selecting the appropriate equation(s), and plugging in the values to solve for the unknown variable.

What are the key equations of motion used in kinematics problems?

The key equations of motion are displacement (Δx = v0t + ½at2), velocity (v = v0 + at), and acceleration (a = Δv/t).

How do I know which equation to use in a kinematics problem?

The equation(s) to use depends on the given information and what you are solving for. If the initial or final velocity is not given, you can use the equation that does not have that variable. If you have three out of the four variables, you can use any of the equations to solve for the fourth variable.

What are some common mistakes to avoid when solving kinematics problems?

Some common mistakes to avoid include forgetting to convert units, using the wrong equation, and not paying attention to the direction of motion. It is also important to double-check your calculations and make sure they are consistent with the given information.

How can I check if my answer is reasonable for a kinematics problem?

You can check the reasonableness of your answer by using estimation and comparing it to the given information. For example, if your calculated final velocity is much greater than the speed of light, then you know there is an error in your calculations. Additionally, you can also check if your answer has the correct units and is in the correct range of values.

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