How do power and velocity affect acceleration in a moving car?

[magiclink]

Hi! I'm extremely confused by this problem. A car of mass 1200 kg starts from rest and travels along a horizontal straight road. The engine of the car develops a constant power of 54 kW. All the energy produced by the engine goes into increasing the kinetic energy of the car.

The questions about this scenario go as follows (at time=5)
What is the instantaneous velocity?
I thought simply using Kinetic energy formula I could figure out the speed= 2.1m/s (roughly)

Now what confuses me is the next question. It asks to find the instantaneous acceleration at t=5. I figured that P=f*v would be a good choice. But other than that I'm at a loss. the formula doesn't quite make sense to me, although i know it's basically f*d/t. If power is constant however, would that mean the force acting on the car is also constant? And wouldn't this mean the acceleration is constant too? Then why does the instantaneous acceleration change at different velocities? Could someone explain the ins and outs of this formula? I cannot wrap my head around it.

P=F*v
(mv^2)/2

 

[Dickfore]

Since the engine develops constant power, the total work done after time t is:
$$W = P \cdot t$$
It says that all work is expended on increasing the kinetic energy of the car. Since the car starts from rest, the initial kinetic energy is zero, and, therefore:
$$T = W$$
where T stands for the kinetic energy of the car. Using the expression for the kinetic energy, you can solve for the instantaneous speed of the car. What I get is ten times bigger than your quoted result.

 

[Dickfore]

As for the instantaneous acceleration, you are right, knowing the instantaneous velocity, and instantaneous (constant in this case) power output, enables you to find the instantaneous force delivered by the engine:
$$F = \frac{P}{v}$$
Then, use Second Newton's Law to find the acceleration.

 

[cepheid]

If power is constant however, would that mean the force acting on the car is also constant?

Nope. The force on the car cannot be constant. Think about this:

F*v = const.

v is increasing

Therefore, what must be happening to F?

 

[magiclink]

Hi! Thankyou so much for the surprisingly quick response! utterly delighted ^^

Okay. I understand the notion that if F*v is a constant then force must be getting smaller as velocity increases. But why is this? intuitively i feel like the force on the car must just be constant throughout. Is the notion that power being constant and therefore force being constant wrong? I see how the equation works but it just doesn't seem to click when i don't look at the formula objectively.

How would the acceleration vs time look for such a scenario? I would've thought it would just be a straight horizontal line?

 

[rcgldr]

Okay. I understand the notion that if F*v is a constant then force must be getting smaller as velocity increases. But why is this?

Imagine the car has a lossless continously variable transmision (CVT). The engine rpms remain constant with the engine producing maximum power, but the gear ratio (engine rpm) / (tire rpm) decreases with speed, so that amount of torque going to the rear tires decreases with speed.

How would the acceleration vs time look for such a scenario?

Similar to a hyperbola, because F = P/v, and power is constant. A t=0, acceleration is infinite, ignoring the fact that real tires would not have infinite friction. The acceleration is initially infinite just for an instant, so it doesn't result in infinite velocity.

Getting back to the original problem, how could you relate the kinetic energy of the car to velocity and to the power from the engine?