A 12.0 -g rifle bullet is fired with a speed of 380 m / s into a ballistic pendulum with maxs 6.

A 12.0 -g rifle bullet is fired with a speed of 380 m / s...

Key Concepts

Ballistic Pendulum

A ballistic pendulum is an experimental apparatus used to measure the speed of a projectile. It involves a projectile embedding into a pendulum bob, and the subsequent rise in the pendulum due to the conversion of kinetic energy to gravitational potential energy is analyzed using both momentum and energy conservation principles.

Kinetic Energy

Kinetic energy is the energy associated with the motion of an object and is given by the equation ½mv². In the problem, calculating the bullet's initial kinetic energy before it collides with the pendulum, as well as the kinetic energy of the combined system immediately after the impact, is essential.

Conservation of Energy

In an isolated system, energy is neither created nor destroyed but can transform from one form to another. For a ballistic pendulum, once the collision occurs, the kinetic energy of the bullet-pendulum system is converted into gravitational potential energy as the pendulum rises. This principle allows us to relate the post-collision speed to the maximum vertical height reached.

Conservation of Momentum

This concept states that the total linear momentum of an isolated system remains constant if no external forces act on it. In the context of a ballistic pendulum, momentum conservation is used during the collision between the bullet and the pendulum, allowing for the calculation of the combined velocity immediately after impact.

Inelastic Collision

An inelastic collision is one in which the colliding bodies stick together after the collision, resulting in a loss of kinetic energy that is transformed into other forms of energy such as heat. The ballistic pendulum problem is an example of an inelastic collision, where the bullet embeds in the pendulum, and while momentum is conserved, kinetic energy is not.

Transcript

00:01 In this question, a bullet of 12 grams is fired at a velocity of 380 meters per second in the direction of a ballistic pendulum with a mass of 6 kilograms and a length of 70 centimeters.

00:13 Then, that bullet will penetrate in the ballistic pendulum, and after that, the pendulum will swing, reaching some height, age.

00:22 Then, in the first item of this question, we have to determine what is that maximum height age.

00:28 For that, we can use energy conservation, but be careful, because energy conservation holds in between two and three only, because during this collision, some energy is lost in this process.

00:43 Therefore, energy is not conserved in between one and two.

00:47 It is not destroyed also.

00:49 It actually becomes heat.

00:51 Therefore, in between one and two, some energy is transformed into heat, so we can't use the principle of energy.

00:58 Conservation between processes 1 and 2.

01:01 Therefore, what we can do is the following.

01:04 In between 2 and 3, energy is conserved.

01:07 It means that the total energy e in situation number 2 must be equal to the total energy e in situation number 3.

01:17 In the situation number 2, we only have the kinetic energy of the ballistic pendulum and the bullet.

01:25 This is provided that we choose a reference frame, such that our horizontal axis crosses right here where i'm pointing.

01:34 So, in fact, this position means that the y -coordinate or the height in some sense is equal to zero.

01:44 Therefore, there is no gravitational potential energy given that we choose this reference frame, which is the frame that i'm choosing.

01:53 Therefore, in situation number two, the energy is given only by that kinetic.

01:58 Energy.

01:59 So we have the mass of that bullet plus the mass of the pendulum and this is multiplied by the velocity of the set v squared and then divided by two.

02:13 That velocity v squared is the velocity with which both the pendulum and the bullet are moving after that collision.

02:21 Then we read situation number three.

02:25 In situation number three, the pendulum will swing and then it will stop at some height and swing back.

02:32 At that maximum height, the pendulum has a zero velocity.

02:35 Therefore, there is no kinetic energy when the pendulum reaches the maximum height.

02:40 Then, all the energy in situation number three is gravitational potential energy.

02:45 And the equation for a gravitational potential energy is the following.

02:49 It is given by the mass, actually the mass of the bullet, plus the mass of the pendulum, because both are at that height age, times g, which is the acceleration of gravity, times the height age.

03:04 Now, we solve this equation for the height age.

03:07 But before, let us simplify out a common factor.

03:10 We can divide both sides of this equation by that factor, so we end up with the following relation.

03:17 V squared over 2 is equal to g times h.

03:21 Then, h, the maximum height, is equals to v squared, divided by 2 times g.

03:28 Okay, but we don't know what is that velocity v.

03:31 How can we proceed? well, we can actually discover that velocity v by using the principle of momentum conservation in between situations 1 and 2...

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