A 0.450-kg hammer is moving horizontally at 7.00 m/s when it...

A 0.450-kg hammer is moving horizontally at 7.00 m/s when it strikes a nail and comes to rest after driving the nail $1.00 \mathrm{cm}$ into a board. Assume constant acceleration of the hammer-nail pair. a. Calculate the duration of the impact. b. What was the average force exerted on the nail?

A 0.450-kg hammer is moving horizontally at 7.00 m/s when it strikes a nail and comes to rest after driving the nail $1.00 \mathrm{cm}$ into a board. Assume constant acceleration of the hammer-nail pair.
a. Calculate the duration of the impact.
b. What was the average force exerted on the nail?

Key Concepts

Kinematics of Uniformly Accelerated Motion

This concept involves using the equations of motion that connect displacement, initial and final velocities, time, and acceleration under the assumption of constant acceleration. It is fundamental for analyzing situations where an object decelerates or accelerates over a known distance or time interval, allowing one to compute any variable when the others are known.

Impulse-Momentum Theorem

This theorem states that the impulse experienced by an object, which is the product of the average force and the duration of impact, is equal to the change in momentum of the object. It is particularly useful in collision problems to relate the force applied over a short time period to the velocity change of the object.

Work-Energy Principle

The work-energy principle explains that the work done by a force acting on an object results in a change in the object's kinetic energy. In collision scenarios, this principle can help relate the distance over which a force acts to the amount of kinetic energy dissipated, linking force, displacement, and the changes in motion.

Transcript

00:01 Okay, so we got a hammer striking a nail, right? so let's draw a picture of this here.

00:06 We got a wall.

00:09 You have a nail, right? and this nail is driven in one centimeter.

00:16 So our distance is 0 .01 meters that it's pushed in, right? that's quite a bit to push a nail in, but let's just go for it.

00:26 And the way it works is that we've got this hammer.

00:29 I'm just going to draw the hammer here.

00:33 Okay, and the hammer has a mass of 0 .4.

00:36 Five kilograms right and its initial velocity is that it's going seven meters per second right and of course the nail stops it so it's flying along at seven meters per second it strikes the nail drives the nail point zero one meters in and they say a calculate the duration of impact right which would be the time it takes to stop right and then what was the average force be let's find the force exerted on the nail, right? which, of course, would be the opposite of the force that the nail exerts on the hammerhead.

01:19 So we'll just find the magnitude of this.

01:22 Now, the trick in this is what we want to do is we want to do force times time.

01:28 This is, of course, our impulse, right? i wasn't going to explain it this way, but i just thought, what the heck, you know? why not explain it this way? you know? i can't hear you laughing.

01:41 All right.

01:42 And then this is our impulses are our change.

01:44 Momentum.

01:45 That's what this is.

01:46 Right.

01:47 Okay.

01:47 So the trick here, though, is that we don't know the time it took to stop, right? so we've got to think about this.

01:54 I guess, you know, here's what we could do, right? isn't our time going to be this? let's do a quick kinematic calculation here, right? time is just going to be distance divided by average velocity, right? so time would be the 0 .01 meters that it travels...

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