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

• Ambidext
In summary: Thanks.In summary, the car with mass 1000 kg and applying a power of 100 kW for 2 seconds will reach a velocity of 100 km/h.
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.

## The Attempt at a Solution

The question hints:
that I need to solve an ordinary differential equation, using a trial solution v(t) = A t1/2, 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.

Ambidext said:

## 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.

## The Attempt at a Solution

The question hints:
that I need to solve an ordinary differential equation, using a trial solution v(t) = A t1/2, 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?

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 :)

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.

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

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]

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?

Yes, what differential equation??

Power is constant.

dE/dt = power = constant.

The kinetic energy E=?

Ambidext said:
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)..

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'

Phrak said:
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.

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!

OK. good to hear it.

## 1. How is power P related to the acceleration of a car?

Power P is directly proportional to the acceleration of a car. This means that the higher the power P, the faster the car will accelerate.

## 2. What are some factors that can affect the acceleration of a car?

The acceleration of a car can be affected by various factors such as the weight of the car, the engine power, the aerodynamics of the car, and the road conditions.

## 3. What are the results of increasing power P in a car?

Increasing power P in a car will result in a faster acceleration, allowing the car to reach higher speeds in a shorter amount of time. However, it can also lead to increased fuel consumption and more wear and tear on the engine.

## 4. How long does it take for a car to accelerate with a certain power P?

The time it takes for a car to accelerate with a certain power P depends on various factors such as the weight of the car, the power of the engine, and the road conditions. Generally, a car with a higher power P will accelerate faster and reach its top speed in a shorter amount of time.

## 5. Can a car's acceleration be improved by increasing power P?

Yes, a car's acceleration can be improved by increasing power P. However, there are limits to how much power P can be added before it becomes detrimental to the car's performance and efficiency. It is important to consider other factors such as weight and aerodynamics when trying to improve a car's acceleration.

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