Accelerating a Car with Power P: Results & How Long?

[Ambidext]

Homework Statement

A car with mass m having an automatic gear box accelerates from rest with a constant power P.
Determine and sketch the position x(t), the velocity v(t), and the acceleration a(t).
How long does a car with mass m = 1000 kg applying a power P of 100 kW to reach a velocity of 100 km/h.

Homework Equations

The Attempt at a Solution

The question hints:
that I need to solve an ordinary differential equation, using a trial solution v(t) = A t^1/2, where A is a constant.

I can sketch the graphs but as my maths is quite terrible for a science student, I have problems understanding the hint and how to apply it.

 

[berkeman]

Homework Statement

A car with mass m having an automatic gear box accelerates from rest with a constant power P.
Determine and sketch the position x(t), the velocity v(t), and the acceleration a(t).
How long does a car with mass m = 1000 kg applying a power P of 100 kW to reach a velocity of 100 km/h.

Homework Equations

The Attempt at a Solution

The question hints:
that I need to solve an ordinary differential equation, using a trial solution v(t) = A t^1/2, where A is a constant.

I can sketch the graphs but as my maths is quite terrible for a science student, I have problems understanding the hint and how to apply it.

Well, power is energy per time or work per time. And work is force multiplied by distance. So what does having constant power applied mean in the equations of motion? Can you start to set up the equations for this motion?

 

[Theaumasch]

There is a relationship between a and v. Use that and you'll get a nonlinear differential equation, which you can solve by rewriting x"x', where ' implies differentiation by t.
Good luck :)

 

[Ambidext]

I started off by using the definition of power,

P = m (dv/dt) v.
dv/dt = P/(mv)

Equations of kinematics I use are:

s = ut + 0.5at²
v = u + at

since I think I would want a and t.

From the ODE onwards, I am stuck.

 

[Theaumasch]

You can't use the standard equation,because a isn't constant. What's the derivative of v²?

 

[The legend]

try using the hint given by berkeman..
P = W*t
W = F.d
P=F.d*t
P=m*a*d*t

now... a little of thought will give you the relation between the given quantities.
[HINT: write out dimensions and use 'em to get the relation]

 

[Ambidext]

We're asked to solve using differential equations. That's the tough part, I think.

I can't make out a different equation with:

P=F.d*t
P=m*a*d*t

though I have no problems getting there. How do I link these concepts to differential equations?

 

[Phrak]

Yes, what differential equation??

Power is constant.

dE/dt = power = constant.

The kinetic energy E=?

 

[The legend]

We're asked to solve using differential equations. That's the tough part, I think.

I can't make out a different equation with:

P=F.d*t
P=m*a*d*t

though I have no problems getting there. How do I link these concepts to differential equations?

sorry, i didnt know that..(i thought the question just hinted you to solve that way)..
You can follow Phrak's advice.

 

[Phrak]

If you must do it the hard way, you had it in post #4.

P = m(dv/dt)v = mv'v or P = mx''x'

 

[Ambidext]

Yes, what differential equation??

Power is constant.

dE/dt = power = constant.

The kinetic energy E=?

If dE/dt = constant, E must be a straight line starting from 0, ie E = k. t where k is a constant.

 

[Ambidext]

Okay I got this question right already. I was just very confused with the phrasing and requirements. The differential equations are meant for sketching the graphs of v(t), x(t) and a(t).

t can be easily calculated with 2 steps. Sorry about my confusion!

 

[Phrak]

OK. good to hear it.