In this lecture the following are introduced:
• Power as the rate of doing work
• Engine power and speed
• Drag
• Efficiency
• Conservation of Momentum
• Inelastic and Elastic Collisions
• Explosions
• Rockets

Definition of Power

Power is broadly, the rate at which energy is delivered, or narrowly, the rate at which work is done.

The unit of Power is the Watt, which is 1 Joule per second.

Human power

The following table gives the typical power (in Watts) consumed by a man with 1.75m² surface area, height 1.75m, and mass 76kg.

Info from: P.Webb in J.F.Parker & V.R.West (Eds) Bioastronautics Data Book.

Note: In the Harvard Step test one steps up and down a 0.4m step 30 times/minute for 5 minutes.

Example P1

A man uses 300W when walking. A meat pie provides 0.5MJ of useful energy. Find how far the man has to walk to use up the energy from the pie.

Engine power and speed

To find the instantaneous output power from an engine, replace work with force and distance.

As Power is a scalar (in Watt), this involves the scalar product of two vectors.

Example P2

An elevator has a weight of 5000N when empty and has no counterweight. It is required to carry a maximum load of 20 passengers (700N each) from the ground floor to the 50^th floor in 40s. The distance between floors is 3.52m. Find the minimum constant power needed.

Drag

When a car or boat is accelerated from rest at full throttle, it will not increase speed without limit, it will eventually settle to a constant speed. This speed is the terminal speed for the object, and it depends on the available power from the engine and the aerodynamic drag. The drag force at low speeds is proportional to the speed. At higher speeds the drag is proportional to the square of the speed, then the 4^th power etc.

For a car at typical speeds the drag force, D, is given by:    

where
      • C_D is the drag coefficient produced by the shape of the car.
      • A is the "front-on" cross-sectional area.
      • ρ is the density of the fluid.
      • v is the speed of the car.

For most cars C_D lies between 0.2 and 0.5.

The resultant force on a car, when drag is considered, is given by:

The resultant initial acceleration causes the speed to increase from zero. This will increase the speed initially but since the acceleration is reduced by speed the acceleration will decrease with time and the speed will not increase continually. Equilibrium is achieved when the drag force equals the force the engine can provide and the acceleration goes to zero. The result is a constant speed called the terminal speed.

Terminal speeds for objects falling in air

From: Peter J. Brancazio, Sport Science.

Efficiency

In every energy exchange there is some energy lost (usually in the form of heat).
Sometimes this is desirable, for example the brakes on a car turn the kinetic energy of the car (linear motion) into heat in the brake pads or drums (random motion).
In other situations it sets limits, for example, the waste heat when a battery converts chemical energy into electrical energy determines the maximum power that you can extract from it.
In yet other situations, it is undesirable and just increases costs.
The power one gets out of a machine is then less than the power provided to the machine.

The efficiency of a machine is measured by the percentage of useful power out to the total power in.

Example P3

A 50kg woman climbs a mountain 3000m high. A Kilogram of fat supplies about 38 MJ. The woman can convert fat into mechanical energy with 20% efficiency. Assuming the fat supplies all the energy to do work against gravity. Find
(a) the work done against gravity, in climbing the mountain, and
(b) the fat consumed by the mechanical energy against gravity.
Note: this neglects the basal metabolic rate of a human body in maintaining temperature.

The mechanical energy needed would be 7.35MJ and this could be provided by 0.19kg of fat.

Newton's first law says:

Bodies remain in a state of rest, or of constant speed in a straight line, unless compelled to change by a push or a pull.

re-phrasing this gives the following equivalent statement:
In the absence of external forces on a system the momentum of the system stays unchanged (i.e. it is conserved).

Collisions

When objects collide they exert equal action-reaction forces on each as they touch and deform.

Considering both masses above as "the system" and neglecting any external forces, such as gravity or drag, there are no forces when the objects do not touch and only internal forces act when they do touch. As only forces external to the system will cause the centre of mass to move, the internal forces during collision don't cause the centre of mass of the system to change.

The momentum of the centre of mass of the system before the collision = The momentum of the centre of mass of the system after the collision

Inelastic and Elastic Collisions

In perfectly inelastic collisions the objects stick together; and to do this, energy is required to change their shape.

Kinetic Energy is not conserved in perfectly inelastic collisions.

In perfectly elastic collisions the objects bounce off and regain their shape perfectly.

Kinetic Energy is conserved in completely elastic collisions.

Example P4

A police constable of mass 65kg (including a bullet-proof vest) is at rest on an ice rink (i.e. no external frictional force) and aiming at an armed offender.   The offender (a lapsed member of the NRA) shoots first, firing right to left (negative x direction).   The constable's vest absorbs the bullet without any body penetration.   The muzzle velocity of the bullet is 320 m.s^-1 and its mass is 50 g. Assume no decrease in bullet speed before impact.   Find the final velocity of the constable and bullet assuming all motion is in a straight line in the x direction.

The negative sign indicates that the constable moves right to left, ie, same direction as the offender's bullet.

Example P5

A car with mass 1500 kg travelling North at 92 km.hr^-1 is struck by another car of mass 2000 kg travelling East at 60 km.hr^-1. The cars lock together on collision. Assuming no external forces act, find the resulting velocity of the combined cars just after collision.

pbefore = pafter m1u1 + m2u2 = (m1 + m2)v
From Pythagoras' theorem, the lengths are related as	From the definition of the tangent The angle is (90-49)=41° clockwise from North.

The final velocity is 52 km.h^-1 at 41° clockwise from the North.

Example P6

A billiard ball of mass 0.5 kg travelling with speed 8 m.s^-1 hits the cushion at an angle of incidence of 41.4° and rebounds without loss of speed so that the angle of reflection equals the angle of incidence. The collision takes 0.2 ms. Find
(a) the change in the momentum of the ball, and
(b) the force of the cushion on the ball.

The change in momentum is the second momentum minus the first, i.e. &Deltap = pafter - pbefore, using vector addition, this is &Deltap = pafter + (-pbefore), as shown in the diagram below

Using trigonometry on the isosceles triangle, by dropping a perpendicular from the apex The change in momemtum is 6 kg.m.s-1 at right angles away from the wall. The force is in the same direction as the change in momentum

Explosions

In an explosion, parts of the initial object exert forces on other parts and so fly apart from each other. As in collisions, only internal forces are acting on the centre of mass of the system.

Example P7

An 80 kg man is standing on a 6.5 kg stationary sled which is on a frictionless ice surface. The man jumps horizontally off the back of the sled with a velocity of 3.5 m.s^-1 due South. Find the sled's velocity immediately after the man jumps off.

Take South as positive.

The sled travels North, very quickly!

Example P8

A radioactive nucleus, initially at rest, decays by emitting an electron and a neutrino at right angles to one another. In one such decay the momentum of the electron is 1.2 x 10^-22 kg.m.s^-1 and that of the neutrino is 6.4 x 10^-23 kg.m.s^-1. The mass of the residual nucleus is 5.8 x 10^-26 kg. Find the velocity of the recoiling nucleus.

Let the neutrino move in the positive x direction, and the electron move in the positive y direction.

Since the initial momentum is zero, the momenta must add up to zero, i.e. in vector terms, they will form a closed polygon. In this case it will be a right angled triangle.

Multiplying all sides by 10²⁶ will simplify the numbers.

From Pythagoras' theorem.

The nucleus moves in a direction which is 152° anticlockwise from the electron and 118° clockwise from the neutrino.

Rockets

Inside a rocket, the fuel explodes and there is an unbalanced force on the fuel because the rocket end is closed and the exhaust end is open. The forces of the exhaust gases on the rocket and the rocket on the exhaust gases are internal forces to the system of rocket and gases. The resultant external force on the rocket and gases is then zero and momentum is conserved. Since this does not depend on any outside influence rockets can work in the vacuum of space.
Now the mass of the rocket varies in time as gas is expelled, so Newton's law of motion takes on a different form as shown below, with the resultant external force on the system being zero.

The product of mass and acceleration is a force in the opposite direction to the velocity of the exhaust gases and is called the thrust. The thrust then depends on the relative speed of the exhaust gases and the amount of fuel per second which is thrown backwards.

When the velocity of the exhaust gases and mass expelled per time remains constant, the thrust will be constant but the acceleration of the rocket will increase as the rocket's mass decreases.

Summarising:

In the absence of external forces on a system the momentum of the system is conserved.

Power is the rate at which energy is delivered, or the rate at which work is done. The unit of Power is the Watt
The output power from an engine is the scalar product of force and velocity
The drag force on a car at typical speeds depends on the square of the speed.
The efficiency of a machine measures the percentage of useful power out to the total power in.

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