A toy cannon uses a spring to project a 5.30 -g soft rubber ball. The spring is originally compr

step 1 In this problem, we're going to talk about the work energy theory. So what we need to remember i

A toy cannon uses a spring to project a 5.30 -g soft rubber...

Key Concepts

Energy Conversion

Energy conversion refers to the process by which one form of energy is transformed into another, such as the conversion of potential energy stored in a spring into kinetic energy. In many mechanics problems, including those involving friction, understanding the pathways and losses in energy conversion is crucial for precisely calculating the system’s behavior.

Kinetic Energy

Kinetic energy is the energy that an object possesses due to its motion, expressed by the formula KE = ½ m v². It quantifies the final energy state of moving objects after energy transformations and losses, and is central to determining speeds in dynamic systems.

Friction as a Non-Conservative Force

Friction is a non-conservative force, meaning that it dissipates mechanical energy (often as heat) during motion. When analyzing systems in motion, accounting for the work done by friction is essential because it reduces the total amount of energy available for conversion into kinetic energy.

Potential Energy in Springs

A compressed or stretched spring stores potential energy that can be calculated using the formula PE = ½ k x². This energy is available to be converted into kinetic energy when the spring is released, and it is a key concept in understanding how force constants and displacement determine the energy output of spring-powered devices.

Hooke’s Law

Hooke’s Law describes the relationship between the force applied to a spring and the displacement it produces, stating that the force is directly proportional to the displacement from equilibrium (F = kx). This principle is fundamental when analyzing systems where a spring is used to store and release energy.

Work-Energy Principle

The work-energy principle states that the work done on an object is equal to its change in kinetic energy. In problems involving energy conversion, such as a spring firing a projectile, this principle allows one to equate the energy provided by the spring (minus any energy losses) to the kinetic energy gained by the object.

Transcript

00:01 In this problem, we're going to talk about the work energy theory.

00:03 So what we need to remember is that the work done on a system is equal to the change in energy of that system.

00:10 So what we have in our problem is a toy cannon that is used to shoot rubber balls.

00:19 And the mass of the rubber ball is 5 .3 grams.

00:24 And inside the cannon there is a spring that is compressed by five centimeters with a spring constant of eight has a constant decay of eight newtons per meter and then the cannon is fired and the ball moves a distance of 15 centimeters inside the cannon and a force of 0 .032 newtons this is a friction force acts on the rubber ball and our going question a is to find what is the velocity or it is the speed of the ball as it exits the cannon so basically the setup is the falling considering that this green part here is the cannon and then we have the spring right here, which is five centimeters long.

01:28 It's compressed all the way.

01:32 It's five centimeters.

01:36 And then the ball is fired.

01:40 Now notice that the force, i'm sorry, the work that is exerted over the ball is minus f times d.

01:48 And the minus is here because the friction force will decrease the energy of the system instead of increasing it.

01:57 And this is equal to the variation in energy.

02:00 Now, the initial energy, i'm sorry, the final energy of the system is just one half of mv squared.

02:07 This is the kinetic energy of the system, minus one half of k times five, we're going to call it delta x squared, because this is the initial energy of the system.

02:25 The system has only the elastic potential energy.

02:30 Okay, then i'm going to find v.

02:33 I'm going to isolate it.

02:34 So notice that v is equal to the square root of k over m delta x squared minus 2f over md.

02:46 This means that v is equal to the square root of 8 newtons per meter divided by 0 .0053 kilograms times 0 .05 meters squared minus 2 times 0 .032 neutrons divided by 0 .0053 kilograms times d that's your point 15 meters.

03:16 So v is equal to 1 point, i'm sorry, that out to 1 .4 meters.

03:26 Then in question b, we have to find at what point the velocity of the ball is maximum.

03:41 Notice that the velocity will be maximum when the acceleration is zero because initially, well i'm going to plot here a graph, qualitative graph of the acceleration as a function of time, actually as a function of x.

04:01 Initially, the acceleration grows because...

04:06 Actually, i'm sorry, the acceleration doesn't grow.

04:09 Initially, the acceleration is a positive number, and then it starts decreasing due to the fact that the spring becomes more relaxed and also due to the fact that there is dissipative force, the friction force.

04:23 And then the acceleration starts to decrease until it becomes zero and then continues to decrease.

04:34 The velocity will increase until the acceleration is equal to zero.

04:39 When the acceleration is equal to zero, the velocity will start to decrease.

04:45 So what we have to do is to find at what point the acceleration is equal to zero.

04:51 That amounts to find at what point the total force is equal to zero...

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