# If one object, moving at a constant speed collides with another object - which is at rest - how much force has been applied?

This is one question for which an answer doesn't really make intuitive sense. In elementary school, we learn F=ma, but where's the "a" in a collision such as this?

Edit: Plus, there has to be a force involved - otherwise, why worry about rear-ending the guy in front of you? His car won't be damaged.

• It doesn't make sense because you are missing an important parameter of the collision: how long does it last? That, of course, depends on the elasticity of the two objects. Imagine placing an ideal massless spring between two perfectly hard balls. The moving ball will compress the spring, which will then exert an accelerating force on the initially resting ball. The softer the spring, the longer the compression will last and the longer it will take to accelerate the second ball... at a smaller acceleration. Jun 11, 2015 at 7:19
• There is no answer, because it depends on how stiff they are. If they are tennis balls (springy) the force will be less than if they are steel (hard). Jun 11, 2015 at 7:19

In a collision it's often the case that it's hard to measure exactly how long the collision lasts and exactly how the force between the objects changes during the collision. Squishy objects like nerf balls will collide relatively slowly while hard objects like billard balls will have a short collision time.

However there is a well defined quantity called impulse that we tend to use in describing collisions. To see how this works suppose we measure the force as a function of time during a collision of two nerf balls, then we might get a graph like:

The force is zero until the two balls touch, then it rises as the balls squish each other. As the balls start to move apart again the force decreases and goes back to zero when the balls separate. So the force, and therefore the acceleration, during the collision has a complicated variation with time.

However if we measure the area under our force-time graph (the orange shaded area) the result is a quantity called the impulse, and this is equal to the change in momentum of the balls. Mathematically we get this by integrating the force-time curve:

$$I = \int F(t) dt$$

Anyhow, the change of momentum is easy to measure because we just have to measure the velocity of the balls before and after the collision. Then once we know the change in momentum, $\Delta p$, we can say:

$$\Delta p = \langle F \rangle t$$

where $t$ is the time the collision lasts, and $\langle F \rangle$ is an average force. This allows you to get some idea of the average forces involved if you know roughly how long the collision lasts.

I think what you would do here is find the momentum. Momentum = Mass x Velocity. so lets say the objects speed is 10 mph, and the weight is 15 lbs. When the object in motion hits the resting object, it will deliver 150 lbs of force.

If the object in motion is going 30 mph and weighs 2000 lbs (lets say a car) then the force it will deliver upon impact with the resting object would be 60'000 lbs. I hope this helps.

• The force is the rate of change of momentum, so you can't calculate the force just from the momentum change. You need to know how rapidly it changes. Jun 11, 2015 at 9:07
• I see what you mean. would you mind explaining this a little more? Jun 12, 2015 at 9:09
• I think this is pretty well covered in my answer Jun 12, 2015 at 10:10

The two vehicles experience a force of the same magnitude due to Newton's third law:

If object AA exerts a force FABFAB on object BB, then object BB will exert a fore FBAFBA on object AA and FBA=−FAB FBA=−FAB However, what you're probably thinking about is that motion of the car is more drastically affected by the collision. This can be explained by Newton's second law. Let's say the truck has mass MM and the car has mass mm. If the magnitude of the force that both vehicles experience is FF, then the magnitudes of their respective accelerations are atruck=FM,acar=Fm atruck=FM,acar=Fm and combining these we get atruckacar=mM atruckacar=mM So if the mass of the car is a lot less than the mass of the truck, then the acceleration of the truck is much smaller than the acceleration of the car, and if you were to watch the collision, the truck would pretty much seem like it's motion was unaffected, but the car's motion will change quite a bit.