Impulse, Momentum and Energy

What is an elastic collision?

A collision in which the two objects bounce apart after colliding.

What is a perfectly inelastic collision?

A collision in which the two objects stick together and move with a final common velocity.

What is the spring constant?

k; a property of a spring that measures the stiffness or stretchiness of a spring. It is also the slope of a force-vs.-displacement graph.

Under what two conditions is mechanical energy conserved?

1. The system is isolated. 2. There is no friction or drag that would transform kinetic or potential energy into thermal energy.

A baseball bat is pushed with a force F. At which location will the force produce an acceleration of the center of mass with the largest magnitude? Explain. 1) At butt of the bat. 2) At center of mass. 3) At head of the bat. 4) Any location will produce the same magnitude of acceleration of the center of mass.

4; the external force is equal to the total mass * instantaneous acceleration of the center of mass. It doesn't matter where the force acts with regards to the center of mass acceleration.

A spring-loaded gun shoots a plastic ball with a speed of 4m/s. Id the spring is compressed twice as far, the ball's speed will be...?

8m/s; since Us depends on (Δx)^2, doubling the compression increases Us by a factor of 4. All of the potential energy is converted to kinetic energy, so K increases by a factor of 4. But K depends on v^2, so v only increases by a factor of 2.

What is a conservative force?

A force for which the work done on a particle as it moves from an initial to a final position is independent of the path followed.

What is a nonconservative force?

A force for which the work is not independent of the path.

What is a restoring force?

A force that restores a system to an equilibrium position.

What is an impulsive force?

A large force exerted for a small interval of time.

What is a collision?

A short-duration interaction between two objects.

What is an isolated system?

A system in which the net external forces is zero (i.e. no external forces or for which the external forces are balanced and add to zero).

What is an elastic system?

A system that exhibits restoring forces.

Suppose you have three objects with equal mass and shape A, B, and C. You throw each of these objects in the air at different times. At an instant, object A is just above the ground with a speed = 3v. B is above A, with a speed = 2v. C is above B and A, with a speed = 0. Which object has the largest gravitational potential energy? Explain.

A; since object A is at a height which is greater than either of the two objects, it has the largest value of Ug. Gravitational potential energy depends only on height and mass, not on speed.

What is kinetic energy?

An energy of motion, measured by the product of 1/2mass * velocity^2.

What is potential energy?

An energy of position, measured by the product of mass * gravity * height.

What is thermal energy?

An energy of temperature.

The impulse-momentum theorem states...?

An impulse delivered to a particle changes the particle's momentum (impulse = change in momentum).

As a ball is thrown in the air, what is happening to its kinetic and potential energy over time?

As it rises, the ball loses kinetic energy and gains potential energy. As it falls, the ball loses potential energy but gains kinetic energy. However the sum of both energies remains constant, as the energy of the system is conserved.

What type of force will always increase thermal energy (i.e.never decrease it)?

Dissipative forces always increase the thermal energy.

What does the basic energy model illustrate?

Energy is transformed within a system without loss, the energy of an isolated system is conserved. Also, energy is transferred to (and from) the system by forces acting on the system. These forces are doing work on the system.

For a force perpendicular to motion of an object, its work is...?

Equal to 0. A force everywhere perpendicular to the motion does no work because ΔK = 0.

A box slides along a frictionless U-shaped slope (e.g. a skate ramp). It is released from rest at height Yi on the left side, and slides down the slope moving towards the right side. Will its final height reached on the opposite side be greater than, equal to, or less than the height at which is was released? Explain.

Equal to; mechanical energy is conserved on a frictionless surface. Because both its Kf and Ki are conserved, we know that Uf = Ui, thus Yf = Yi. The final height must match the initial height.

How can you find force from potential energy?

F = the negative slope of the potential energy-vs.-position graph.

The change of the total momentum of an isolated system is equal to the ________________ applied to the system.

Net force; an isolated system does not experience a change in momentum because its momentum is conserved.

How does friction affect the energy in a system?

Friction raises the temperature, so a system with friction transforms kinetic or potential energy into thermal energy.

How do you measure spring force?

Fsp = kΔx, where k is the spring constant.

What does the sign of work represent?

If W is positive, the environment does work on the system and the system's energy increases. If W is negative, the system does work on the environment and the system's energy decreases.

Suppose the entire population of the world gathers in one spot. At a signal, everyone jumps upwards. While all of the people are in the air, what happens to the Earth's momentum? Explain.

If we consider this an isolated system, the Earth will gain momentum in the opposite direction. This is because the Earth will recoil, like a bullet fired from a rifle. Its change in momentum will be equal to and opposite to that of all of the people.

In terms of what is conserved in inelastic and elastic collisions, how do each type compare?

In isolated systems, perfectly inelastic and perfectly elastic collisions both conserve momentum. Perfectly elastic collisions also conserve mechanical energy.

If a ball drops to the earth, will its momentum be conserved?

It depends on the system; if you consider the system as only the ball itself, then momentum is not conserved because it experiences the external force of gravity that changes its momentum. If you consider the system as the ball + the earth, then momentum is conserved because it would be an isolated system.

Suppose a ping-pong ball and a bowling ball are rolling toward you. Both have the same momentum, and you exert the same force to stop each. How does the distance needed to stop each ball from rolling compare? Explain.

It takes a larger distance to stop the ping-pong ball; since the momenta is equal for both balls, the ball with larger velocity will have a larger kinetic energy. Since the ping-pong ball has a smaller inertial mass, it has a larger velocity. This means more work must be done on it in order to stop it. Since work = F*Δx, displacement will be greater for the ping-pong ball.

Mechanical energy is conserved as long as which type of force is not present?

Mechanical energy is conserved if there are no nonconservative forces (ΔEmech = 0, if Wnc = 0).

The sum of total kinetic and potential energy in a system is also called...?

Mechanical energy.

Impulse transfers ____________ to an object.

Momentum

Two particles A and B are moving to the right. Particle A catches up with particle B and collides with it. The particles stick together and continue on with a common final velocity. What can we say about their velocities?

Momentum conservation requires that (mass A + mass B) * the final velocity = (mass of A * velocity of A) + (mass of B * velocity of B). Because velocity of A > velocity of B, we know that the final velocity of both > velocity of B, and the final velocity of both < velocity of A. The collision causes A to slow down and B to speed up.

Can kinetic energy ever have a negative value? Explain.

No; by its definition, kinetic energy cannot have a negative value since its velocity value is squared and mass of an object can never have a negative value. Hence, kinetic energy is concerned with an object's speed (magnitude of velocity), not its actual velocity.

Can elastic potential energy ever have a negative value? Explain.

No; since Δx is squared, the elastic potential energy for a spring is positive for a spring that is either stretched or compressed. It equals 0 when the spring is at its equilibrium length and Δx = 0.

What is a dissipative force?

Nonconservative forces that cause the macroscopic kinetic energy of a system to be "dissipated" as thermal energy (e.g. friction, drag).

Identical constant forces push two identical objects A and B continuously from a starting line to a finish line. Neglect friction. If A is initially at rest and B is initially moving to the right, what can you say about each object's change in momentum? Explain.

Object A will have a larger change in momentum; object A is pushed for a longer time interval than B since its initial velocity = 0. This means that it will have a larger impulse (Favg*Δt) = change in momentum.

What is an explosion?

Occurs when the particles of a system move apart from each other after a brief, intense interaction (i.e. the opposite of a collision).

A potential energy can be associated with which type of force?

Potential energy can be associated with any conservative force.

An explosion in a rigid pipe shoots out three pieces. A 6g piece comes out the right end. A 4g piece comes out the left end with twice the speed of the 6g piece. From which end, left or right, does the third piece emerge? Explain.

Right end; the pieces start at rest so the momentum of the total system = 0. It's an isolated system, so the total momentum after the explosion will also be 0. The 6g piece has a momentum = 6v, while the 4g piece has a momentum of -8v. The combined momentum = -2v. Since momentum must equal 0, the third piece has to have +2v momentum and thus a positive velocity.

Is energy a scalar or vector quantity? Explain.

Scalar; kinetic energy depends on an object's speed rather than its velocity (not concerned with direction of motion.

How could you describe an object that has a large momentum?

Since momentum is equal to mass * velocity, an object with a large momentum could either have a large mass or be moving with a fast velocity.

What is impulse?

The area under the force-vs.-time curve, = (average force * Δt), = Δp.

In terms of potential and kinetic energy, the law of conservation of energy states that...?

The kinetic energy and the potential energy may change as they are transformed back and forth into each other, but their sum will remain constant (Kf + Uf = Ki + Ui).

What is a spring's equilibrium length?

The length of a spring when it is neither pushing nor pulling (Δx = 0).

Consider two carts, of masses m and 2m, at rest on an air track. If you push both carts for 3s with equal force, how will the kinetic energy of the lighter cart compare to the kinetic energy of the cart twice its mass? Explain.

The lighter cart will have a greater kinetic energy than the heavier cart. Since K = 1/2mv^2, the cart with a faster velocity will have a larger kinetic energy. We know that the velocity of the lighter cart must be twice as fast since both have equal momentum.

What is work?

The mechanical transfer of energy to a system via forces.

Suppose rain falls vertically into an open cart rolling along a straight horizontal track with negligible friction. With the result of accumulating water, what can we conclude about the momentum and kinetic energy of the cart? Explain.

The momentum of the cart does not change, and the kinetic energy of the cart decreases. The water falls vertically, which does not change the cart's horizontal momentum. The mass of the cart increases, but the speed is also decreasing simultaneously. This would mean that its kinetic energy would also be decreasing as it slows.

Consider two carts, of masses m and 2m, at rest on an air track. If you push both carts for 3s with equal force, how will the final momentum of the lighter cart compare to the final momentum of the cart twice its mass? Explain.

The momentum will be the same for both carts. This is because Δp = Favg*Δt. Since both receive the same magnitude of force for the same amount of time, they will have the same change in momentum.

A force does work on a particle if...?

The particle is displaced.

What is elastic potential energy?

The product of 1/2k(Δx)^2 (where k = spring constant).

How do you calculate force in terms of momentum?

The rate of change in momentum, or Δp.

Hooke's law states that...?

The restoring force of a spring can be found using: Fsp = -kΔx.

Two balls of equal mass and size are thrown at a wall with equal speeds. One is made of clay, while the other is made of rubber. The rubber ball bounces, while the clay ball sticks. Which ball exerts a larger impulse on the wall? Explain.

The rubber ball exerts a larger impulse because it bounces. Since the clay sticks, its final velocity is equal to 0. The rubber ball rebounds with an equal yet opposite final velocity to its initial. This would mean that it experiences a larger change in momentum, hence the larger impulse.

What determines the sign of work in a system?

The sign of W is determined entirely by the angle θ between the force and displacement Δr.

When does an object have negative momentum?

The sign of an object's momentum when the object has negative velocity, or is moving to the left of the x-axis or moving down on the y-axis.

When does an object have positive momentum?

The sign of an object's momentum when the object has positive velocity, or is moving to the right of the x-axis or moving up on the y-axis.

On a force-vs.-displacement graph, what would a steep linear graph represent (in terms of k) versus one that is not very steep?

The steeper the graph, the larger the slope. Since k = slope of a force-vs.-displacement graph, a steeper graph would have a larger spring constant than one that did not have a steep line.

What is system energy?

The sum of all mechanical energy of the objects plus the thermal energy of the atoms inside the objects (Esys = Emech + Eth = K + U + Eth).

The law of conservation of energy states that...?

The total energy of an isolated system is a constant. The kinetic, potential, and thermal energy within the system can be transformed into each other, but their sum cannot change. Further, the mechanical energy is conserved if the system is both isolated and non-dissipative.

The law of conservation of momentum states...?

The total momentum of an isolated system is a constant. Interactions within the system do not change the system's total momentum.

What is heat?

The transfer of thermal energy in or out of a system.

What is momentum?

The vector quantity found from the product of a particle's mass and velocity.

Can gravitational potential energy ever have a negative value? Explain.

Yes; a negative value for potential energy means that the particle has less potential motion at that point than it does at y = 0. Values for potential energy also vary on how one chooses his or her coordinate system (though the value of ΔU remains constant independent of its axes).

A small child slides down 4 frictionless slides A-D. Each has the same height. Slide A is steep and completely diagonal. Slide B is also completely diagonal, but is less steep than A. Slide C has a concave shape, while slide D has a convex shape. Which slide will yield the largest speed? Explain.

They will all yield the same speed; the child's increase in kinetic energy depends only on the vertical height through which she falls, not the shape of the slide.

A car accelerates from rest. In doing so, the magnitude of the momentum changes by a certain amount. Simultaneously, the Earth's momentum will also change by a certain amount. What can you say about each object's Δp?

They will both have the same amount of change in momentum, since momentum is conserved in this system.

Suppose a ping-pong ball and a bowling ball are rolling toward you. Both have the same momentum, and you exert the same force to stop each. How do the time intervals to stop each ball from rolling compare? Explain.

They will take the same amount of time; since Δp = Favg*Δt, the two balls lose momentum at the same rate. If they both initially have the same momentum, it will take the same amount of time to stop them.

A child at the playground slides down a pole at a constant speed. This is a situation in which...?

U -> Eth, Emech is not conserved but Esys is; constant speed means ΔK = 0, gravitational potential energy is lost, and friction heats up the pole and the child's hands.

A child slides down a playground slide at constant speed. The energy transformation is...?

U -> Eth; constant speed means ΔK = 0. Gravitational potential energy os lost, and friction heats up the slide and the child's pants.

Objects A and C are made of different materials, with different "springiness," but they have the same mass and initially start at rest. A ball thrown at each of the objects at the same speed. When the ball hits object A, the ball ends up at rest. When the ball hits object C, the ball rebounds to the left. Is the velocity of A greater than, equal to, or less than the velocity of C? Explain.

Velocity of A is less than the velocity of C. The ball's momentum is the same in both cases, and the momentum is conserved in both cases. This means that the total momentum is the same after the ball hits either object. The ball that hits C rebounds and has a negative momentum, so C must have a larger momentum than A.

How can you calculate the work done by a constant force?

W = F*Δr, where Δr is the displacement vector.

How do you calculate work?

W = ΔK = area under the force-vs.-position graph.

What is the unit of energy?

We measure energy in joules.

A crane lowers a steel girder into place. The girder moves with a constant speed. Consider the work done by gravity Wg and the work done by the tension in the cable Wt. What do we know about the signs of either object's work?

Wg is positive and Wt is negative; the gravitational force is in the same direction as displacement, so it does positive work. The tension force is opposite to the displacement, so it does negative work.

The work-kinetic energy theorem states...?

When one or more forces act on a particle as it is displaced from an initial position to a final position, the net work done on the particle by these forces causes the particle's kinetic energy to change by ΔK = Wnet.