Two cars travelling towards each other with the same velocity and different mass. They crash. Which one has the largest acceleration?

Two cars are both moving towards each other with the same initial velocity of 50km/hr. One of the cars weighs 500kg, the other weighs 1000kg. They collide and stick together after the collision, travelling with the same final velocity. Which car experiences the largest acceleration?

My teacher said it should be the car with he smaller mass (so the 500kg car) that experiences the largest acceleration. But it made me confused because shouldn't they both have the same acceleration if they both had the same initial and final velocities? Thanks

• If they stuck together, they had the same final velocities. But are you sure they had the same initial velocities? (Remember the difference between speed and velocity.)
– Mike
Commented Sep 18, 2023 at 4:51
• yeah, it's confusing me a lot. this is the exact question: A hot Volkswagen Golf travelling north collided head-on with an urban assault vehicle (otherwise known as an SUV) travelling south. Each vehicle as travelling at 50km/hr before the collision and stuck together after the collision. If the mass of the Golf was half the mass of the obnoxious gas-guzzling SUV, identify the vehicle that experienced the larger acceleration. I changed up the numbers and wrote a shorter version of the same question in my post because I wanted to simplify it a bit, but that's the original question. Commented Sep 18, 2023 at 4:56

The force they exert on each other will be equal and opposite during the collision (Newton's third law of motion). Let's call the magnitude of this force $$F$$, which is the same for both cars.

Then starting with the equation $$F=ma$$ we can rearrange to get $$a=F/m$$.

So if we have a fixed force $$F$$ and a smaller mass $$m$$, we'll get a larger acceleration $$a$$ for the smaller car.

• thank you for the help have a nice day Commented Sep 18, 2023 at 4:59
• the thing is , they have mentioned that both cars stick together at the end so they have the same final velocities and initial velocity is also said to be the same , then I dont know how the force would be equal as a newton's third law pair as F=mdv/dt then dv/dt is same for both cars , then automatically force varies for both cars and it doesn't obey newtons third law There must be an external force that kept this cars intact
– Razz
Commented Sep 18, 2023 at 5:05
• I'm assuming that the initial velocities are equal in magnitude only (and not in direction) since it's a collision. Commented Sep 18, 2023 at 5:10

As the comments have pointed out , the cars dont have the same initial velocity

one is $$50 km/hr$$ and other is $$-50km/hr$$

Let us assume the velocity of small car is positive so and the final velocity to be $$V$$

Then force pairs on small car due to big car is $$F=m\frac{dv}{dt}$$

$$F=m\frac{V-U}{t}$$ and the force on big car would be $$F=M\frac{V+U}{t}$$

Then Let a=$$\frac{V+U}{t}$$ which is the acceleration experienced by the big car and $$a^{'}=\frac{V-U}{t}$$ which is the acceleration experienced by the small car

Since they experience the same force F would be same and hence the $$a^{'}$$ which is the acceleration term for smaller car would be greater for car as $$\frac{F}{m}$$ would be larger for the smaller car

IF it is given that the cars stuck together... then both of the cars will have same acceleration in same direction...( they move like a single object)

Otherwise... The larger car will be accelerated less does not matter what their initial velocites were...

the cars will experience same force according to newton 3rd law... F=ma.... if mass is bigger.. acceleration will be less..for a constant force.. a=F/m

the smaller car will be accelrated more..

its like you and a sumo wrestler run towards each other and collide..... if you are a middleweight then both of you will bounce back in opposite directions... but sumo will just bounce slighyly while you will just bounce far( assuming sumos are fatty heavyweights)