Does it take more energy to bring a car to a halt if it is still accelerating on impact than travelling at constant speed?

Question

So, my physics is quite rusty, been out of varsity for a while.

A friend asked me this and I am still pondering. Here is the scenario:

• 2 Cars are travelling towards a wall, and make impact with the wall at the same speed, e.g. 10m/s.
• Assume the wall is unbreakable and the cars have the same mass, e.g. 500kg
• Car 1 is travelling at a constant speed of 10m/s when it hits the wall, i.e. no acceleration.
• Car 2 is accelerating towards the wall, and hits the wall at 10m/s, i.e. it accelerates all the way until it hits the wall.
• So both cars hit the wall with the same speed, but car 1 has no acceleration, and car 2 has an acceleration.

Does the wall exert more energy to bring car 2 to a stop, because car 1 only has momentum, but car 2 has both momentum and a net force?

This makes sense logically to me but not sure how to explain it in equations.

Answer 1

Imagine that there is no wall, but for some other reason, car 2 instantaneously stops accelerating at the moment when it's speed reaches 10 m/s. Do you think that the passengers would notice?

Engineers have a name for the first derivative of acceleration w.r.t. time (2nd derivative of velocity, 3rd derivative of position.) They call it, jerk, and from the passengers' point of view, it certainly does feel as if they are being jerked around. The passengers feel it because, while they are accelerating, energy is stored in the compression of the springs of their seat cushions and, in internal strain of their body tissues.* Then, when the acceleration is suddenly switched off, that internal, potential energy suddenly is released.

I suspect that your expectation of that extra "jerk" explains your intuition about the crash. To an observer looking from the outside, both cars have exactly the same kinetic energy at the moment when they contact the wall, but the passengers of the accelerated car will also be affected by the release of that stored, internal energy, which will not be felt by the passengers of the constant velocity car.

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* Maybe a bit more complicated than just calling it "strain." See https://en.wikipedia.org/wiki/Jerk_(physics)#Physiological_effects_and_human_perception

Answer 2

"Does it take more energy...", "Does the wall exert more energy..." These phrases do not make much sense. The wall does not lose energy by stopping the car, it does negative work and actually acquires energy through heating. The wall does more negative work stopping the accelerating car (for example, the accelerating car would compress a spring more).

Answer 3

Does the wall exert more energy to bring Car 2 to a stop, because Car 1 only has momentum, but Car 2 has both momentum AND force?

I assume by "exert more energy", you mean does Car 2 absorb more energy than Car 1 when colliding with a perfectly rigid wall. I believe the answer is yes, but only by a relatively tiny amount

An expert in car collision physics (which I am decidedly not) would be the best person to answer this, but consider the following:

Acceleration Force:

To separate the energy absorbed by the acceleration force from the energy absorbed by the average impact force associated with the change in momentum, visualize the car having no velocity and its front bumper in contact with the rigid wall. We then apply a force to the rear of the car equal to the maximum possible accelerating force and measure the compression of the bumper shock absorber spring to calculate the theoretical absorbed energy due to the acceleration force only.

The force that causes the acceleration of the car is the static friction force, $f_s$, of the road acting forward on the drive wheel(s). That force is limited to the maximum possible static friction force, $f_{s-max}$ where

$$f_{s-max}=\mu_{s}N$$

Where $N$ is the normal force of the road acting on the tire and $\mu_s$ is the coefficient of static friction between the road and tire. For simplicity we can take $N$ to equal the weight of the car, $mg$. Thus

$$f_{s-max}=\mu_{s}mg$$

For ordinary (non race car) tires on dry asphalt pavement, $\mu_s\lt 1$. Thus, taking for your 500 kg car and letting $\mu_{s}=1$ the maximum accelerating force becomes

$f_{s-max}=$ 4900 N

If we let the shock absorber spring constant be 410 kN/m $^1$, the compression of the spring would be 0.012 m, and the absorbed energy ($\frac{1}{2}kx^2$) due to the 4900 N force would be 29.3 J

Impact Force:

Now consider the average impact force for stopping the cars in your example. Per the work energy theorem, where the net work done on an object equals its change in kinetic energy.

$$W_{net}=F_{ave}d=\frac{1}{2}mv^2$$ where $F_{ave}$ is the average impact force acting on the car, and $d$ is the stopping distance.

Taking $v$ = 10 m/s (about 22 mph), d= 0.25 m (about 50% of the car crumple zone distance $^2$), and 500 kg for the mass, we obtain an average impact force will be about 25,000 N and absorbed kinetic energy of 6,250 J.

If we were to increase the speed to 27 m/s (about 60 mph) and the average impact force becomes 182,250 N with absorbed energy of 45,562 J.

Hope this helps.

Notes:

1. Value frequently associated with problems solved on internet, though I could not find the source.

2. https://www.sciencedirect.com/topics/engineering/crumple-zone#:~:text=In%20frontal%20and%20rear%20impacts,it%20is%20around%20150%20mm

Answer 4

The reason the car is accelerating is that its wheels are pushing at the ground, and the ground is pushing back - that pushing back is the force which accelerates the car. Imagine that the car is tethered to a helicopter flying above, and while it's accelerating it lifts it off the ground - it'll stop accelerating immediately, the wheels will just turn uselessly. It will still maintain the same speed forward though. Speed (velocity) is something the car "remembers", acceleration is not - it must continue to be applied every instant, with some force.

Now imagine for a second that when both cars hit the wall, you also, at the split second just prior to the crash, switch off the engine of the accelerating car (or, even better, lift it up every so slightly by a helicopter). Then certainly the wall will exert exactly the same energy to stop both cars. The car that was accelerating doesn't "remember" it had been accelerating and is not different in any way than the car that was going at constant speed.

In real life, the same effect (of switching off the engine) will happen because the wall will wreck the car. But it will not happen instantaneously. So for a very short time while the accelerating car has already met the wall, but its engine is still turning the wheels and those push at the ground and the ground pushes back - that force will transmit to the wall and will require the wall to do a little more work (expend a little more energy) than with the constant-speed car.