Momentum and Acceleration

In Physics we've just started on momentum. Here's a problem we did in class.

You may stand in awe at my awesome drawing skills.

Ball is moving at 200 m/s and has a mass of 2kg. Person about to get hit has a mass of 50 kg.
Frictionless surface.
How fast does the person move after the cannon ball transfers it's momentum?

Since p = mv, and momentum is conserved,
2 kg * 200 m/s = 50 kg * x
Solve for x and we conclude the poor sucker moves at 8 m/s.
(I think)

My question is: "How fast does he accelerate?"
I know he doesn't magically go from 0 m/s to 8. Teacher told me you'd need to know more information.

So 1) what else, if anything, do you need to know and 2) how would you work the acceleration out if you had all the required variables?

Momentum is defined as mass times velocity, acceleration is the rate of change of velocity.

To be more accurate, velocity is a vector quantity, it has both magnitude and direction. Momentum is therefore also a vector quantity in the direction of the velocity with magnitude equal to the mass times the magnitude of the velocity:

1) p = mv

Acceleration is also a vector quantity and in the direction of the change in velocity direction and represents the rate of change of velocity:

2) a = dv/dt

Force is defined as the rate of change of momentum, and is therefore also a vector in the direction of the momentum change:

3) F = dp/dt

Substituting 1) in 3) we get:

4) F = m(dv/dt)

And since 2) defines dv/dt as acceleration we get:

5) F = ma
Which, as you know, is one of Newton's laws.


From wikipedia
"When a force is applied to a rigid body it changes the momentum of that body. A small force applied for a long time can produce the same momentum change as a large force applied briefly, because it is the product of the force and the time for which it is applied that is important."

Impulse is force*time, basically.

The formula for force: F= ma
The formula for momentum: p=mv
The formula for impulse: I = delta(F) * time


So, you'd need to know the time.

My question is: "How fast does he accelerate?"
I know he doesn't magically go from 0 m/s to 8. Teacher told me you'd need to know more information.

So 1) what else, if anything, do you need to know and 2) how would you work the acceleration out if you had all the required variables?

1) You don't need more information.
2) IF it's completely frictionless, think of Newton's cradle (except that the balls aren't going to slow to a stop). In your example, the ball will just stop moving and the guy will move in the place of the ball. He goes from 0m/s to 8m/s instantaneously. Now, if there were frictional force, using the normal force, you can calculate for the guy's deceleration.

So, you'd need to know the time.

Why do you need to know the time? He's in a frictionless environment. It doesn't matter if it takes 20 seconds for the ball or 30 seconds for the ball to hit the guy, it's still going to hit the guy with 400N.

Instantaneously...? That's so... xD
If there were friction, drag from the fluid being neglectable, how would you use the normal force to find the acceleration?

What? Normal force IS frictional force...

I was thinking more so of contact force, rather than time to actually reach him [that's clearly irrelevant].

An instantaneous change in speed cannot happen - he isn't hit with the full force all at once. There's a minute amount of time where he feels the tip of the force and more as you go on until you feel the full 400N. In this case, time would be the time between these two 'force feels'.

What? Normal force IS frictional force...

Let's say the floor in the image has a friction coefficient of .25.
How fast does he accelerate?

The same acceleration he would have if he started to move in that direction at a speed of 8 m/s, with m being mball + mperson.

So 400/250 = 1.6 m/s/s?

You have to specify if the type of collision here. Inelastic collision, perfectly inelastic collision, elastic collision.

Since elastic makes it more complicated, inelastic.

If it's inelastic or elastic doesn't matter, what matters is how the force is transferred. The surface friction isn't relevant either in a perfect collision.

The less the person compresses, the closer acceleration is to infinity. If he's an ideal solid, it's infinite (doesn't matter if collision is elastic or inelastic). If it penetrates for 10cm or whatever, then there is time for it to slow down, and speed him up - then you need to find how long it takes from initial contact until the projectile has stopped moving relative to the person, and divide the final speed by that time to get the acceleration.


If it's inelastic the final speed is also slightly different - instead of 50kg mass you have a 52kg mass, so 2*200 = 52*x.

Momentum is defined as mass times velocity, acceleration is the rate of change of velocity.

To be more accurate, velocity is a vector quantity, it has both magnitude and direction. Momentum is therefore also a vector quantity in the direction of the velocity with magnitude equal to the mass times the magnitude of the velocity:

1) p = mv

Acceleration is also a vector quantity and in the direction of the change in velocity direction and represents the rate of change of velocity:

2) a = dv/dt

Force is defined as the rate of change of momentum, and is therefore also a vector in the direction of the momentum change:

3) F = dp/dt

Substituting 1) in 3) we get:

4) F = m(dv/dt)

And since 2) defines dv/dt as acceleration we get:

5) F = ma
Which, as you know, is one of Newton's laws.


From wikipedia
"When a force is applied to a rigid body it changes the momentum of that body. A small force applied for a long time can produce the same momentum change as a large force applied briefly, because it is the product of the force and the time for which it is applied that is important."

Impulse is force*time, basically.

The formula for force: F= ma
The formula for momentum: p=mv
The formula for impulse: I = delta(F) * time


So, you'd need to know the time.

You forgot the source.

Well, let's assume the crazy idea that Hazzy has: that all the momentum is transferred to the human, and that the cannon-ball has 0 momentum afterwards.

A simple answer would then be that the average acceleration is the difference in speed divided by the amount of time. The amount of time here is the amount of time the man and the cannon-ball touch eachother. But since we don't have that, that's the information we need. From what Stereo said, the reason they contact in more than an infinitesimal period of time is because of the compression (and expansion) of the man.

Of course, the average acceleration is not the constant acceleration at any given time t: You could approximate the constant acceleration by using this formula:



Where t = 0 is the start of the touch, and t = 2b is the end of the touch. k is a constant so that



(I don't think this is something you have to say, though.)

Noah

You'd need, for the acceleration at any given time, the force as function of time.