# Ek vs engine power in cars and space

#### vysqn

(1) An idealized engine with constant torque curve means constant acceleration.
So for example if there is no air resistance and no frictions and car is on fixed gear and at 3000Rpm it produces 50Hp so at 6000Rpm will produce 100Hp.
So he doubles the speed , doubles power and got constant acceleration.

(2) So if we look at equation Ek = 1/2mv^2 which means if You wanna accelerate same as before to twice a speed You need 4 times of energy per second(or horsepower per second) So this is what I dont understand because first example shows that car need twice more power at twice speed to same amount of acceleration.
Car with CVT gearbox which acceleration from 0-100km/h in 10s doesnt need 40s to accelerate from 100-200km/h even when air resistance is much bigger (goes quadratic as well)

(3) Higher speeds require more energy for subsequent speed increases, hence at constant power, acceleration goes down. Why we need more energy when speed is increases? Because gravity wanna pull as or what? And why we dont need more energy in space to acceleration at higher speed?

(4) Why it works in cars and doesnt work for example in space? I mean Rocket egine produce same amount of energy per second but acceleration is constant and doesnt decrease when speed increases ?

<< Post edited by Mentors >>

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#### BvU

Homework Helper
Hello vysqn,

I have great difficulty following your logic: Do you propose to accelerate from standstill with a fixed gear ? 0 rpm means zero torque for your ideal engine !
And, supposed you manage to get moving, obviously your acceleration will increase as torque increases, so it will not be constant at all.

Are you aware of the SUVAT equations ? Newton ? If so, try a few of those for your scenarios.

Why we need more energy when speed is increases?
Look at the kinetic energy equation in (2) again !

#### anorlunda

Mentor
Why it works in cars and doesnt work for example in space?
no air resistance and no frictions
It is air resistance and friction that makes us need more power for higher speeds.

A car in space makes no sense (unless you are Elon Musk). The tires need to push against something and friction is what keeps the tires from spinning.

#### vysqn

Look Guys at this explanation from 8:15 to 9:15. It shows that car will need twice power when goes twice fast to get constant acceleration. So why E kinetic shows us that we need 4x of energy to drive twice as fast.
I know about air resistance and frictions but at lower speeds air resistance is very low.
My car which has very flat torque (like table) curve from 2600-6300rpm at 3rd gear for example accelerate at same amount of force from 50-100km/h

#### cjl

(1) An idealized engine with constant torque curve means constant acceleration.
So for example if there is no air resistance and no frictions and car is on fixed gear and at 3000Rpm it produces 50Hp so at 6000Rpm will produce 100Hp.
So he doubles the speed , doubles power and got constant acceleration.
Yes, this is true

(2) So if we look at equation Ek = 1/2mv^2 which means if You wanna accelerate same as before to twice a speed You need 4 times of energy per second(or horsepower per second) So this is what I dont understand because first example shows that car need twice more power at twice speed to same amount of acceleration.
Car with CVT gearbox which acceleration from 0-100km/h in 10s doesnt need 40s to accelerate from 100-200km/h even when air resistance is much bigger (goes quadratic as well)
This is where you're confusing yourself. At 100mph, the car has 4x the energy as at 50mph, and it can maintain the same level of acceleration on twice the horsepower. However, at the same level of acceleration, it takes 2x as long to go from 0-100 as from 0-50, so with constant force, it will have spent twice as long accelerating with twice the average power, for a total of 4x the energy transfer.

Also, a car that can accelerate from 0-100km/h in 10 seconds absolutely will need a very long time to go from 100-200km/h. In the absence of losses, at constant horsepower, it will take 30 seconds to go from 100-200 (since at 200, it has 4x the KE of 100, and so the energy addition from 100-200 is 3x higher than the energy addition from 0-100), but in reality, it will take substantially longer than that. A car that takes 10 seconds to get to 100km/h will be using the vast majority of its power at 200km/h just to fight wind resistance, and I would not be surprised if it took over a minute for such a car to go from 100-200km/h, if it can reach 200 at all.

(3) Higher speeds require more energy for subsequent speed increases, hence at constant power, acceleration goes down. Why we need more energy when speed is increases? Because gravity wanna pull as or what? And why we dont need more energy in space to acceleration at higher speed?
This is unrelated to gravity. Rather, it's based on the way that energy is defined. If you look at it as relating to work (force multiplied by distance), it makes sense that as you're traveling faster, you are covering more distance at the same force level, so more work is done. Alternatively, you can look at what would be required to accelerate a vehicle to twice the speed over the same distance. If you assume a constant acceleration, it will cover the distance in half the time if the final speed is twice as high, so you need to accelerate the vehicle to twice the speed over half the time, resulting in 4x the force applied over the same distance.

(4) Why it works in cars and doesnt work for example in space? I mean Rocket egine produce same amount of energy per second but acceleration is constant and doesnt decrease when speed increases ?
Rocket engines do not do a fixed amount of work on the rocket, nor do they provide a fixed rate of energy transfer. Rather, they provide a fixed thrust. If you want to do an energy analysis, you need to look at the total energy content of the rocket and the exhaust at various rocket speeds, rather than just the rocket alone.

#### jack action

Gold Member
So he doubles the speed , doubles power and got constant acceleration.
So if we look at equation Ek = 1/2mv^2 which means if You wanna accelerate same as before to twice a speed You need 4 times of energy per second(or horsepower per second) So this is what I dont understand because first example shows that car need twice more power at twice speed to same amount of acceleration.
In the first quote, you talk about power, and in the second quote, you talk about energy.

$E_k$ is not «energy per second», it's simply energy. Or «power times second» (or horsepower times second).

From your first quote, the force $F$ is constant and you double the speed $v$. From the definition of power $P$:
$$P=Fv$$
So if $F$ stays the same and $v$ is multiplied by 2, then $P$ is multiplied by 2 as well.

From your second quote $E_k = \frac{1}{2}mv^2$. But we can relate that to power with our first equation$P=Fv$ or $v=\frac{P}{F}$:
$$E_k = \frac{1}{2}m\left(\frac{P}{F}\right)^2$$
If $m$ and $F$ stay the same and $P$ is multiplied by 2, then $E_k$ must be multiplied by 4. So both of your statements agree with each other.

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#### CWatters

Homework Helper
Gold Member
You (vysqn) confuse energy and power. The equation 0.5mv^2 is the Kinetic Energy (KE) of the car not the power. To work out the power you need to know how fast it gained that KE. For example a 1HP (750W) moped will take longer to get to 60mph than a 100HP (75,000kW) car will. Same velocity, different power.

#### anorlunda

Mentor
Somebody should sell attractive walnut plaques to hang on the wall saying:

Energy is to Power as Distance is to Speed

So many people, especially newspaper reporters, have trouble remembering that.

#### OldYat47

Ref. (2) in your post. Think only of the forces involved. In frictionless conditions if the mass is moving at a constant speed then no force is required. Since power = force X velocity power is also zero.
The required force for a given constant acceleration in frictionless conditions is (force = mass X acceleration). Power = force X velocity, so with constant (mass X acceleration), power increases linearly and proportionally to velocity. Under those conditions if velocity doubles, power doubles. If velocity increases 20%, power increases 20%. Force is constant. This is important to note. Under constant force and constant acceleration power changes continually as velocity increases.
Ref. (3) in your post. Any acceleration of any mass in a frictionless domain takes the same amount of force regardless of initial velocity. Aerodynamic drag increases as the square of the velocity so it does take more force to accelerate a mass as the velocity increases in a gas atmosphere. And since power = force X velocity, it takes more power.