# Energy and Power of a Particle

As a force is applied to a particle there is a certain potential for that force to do work on that particle.  Work is defined as the amount of displacement that a particle will undergo in the direction of the applied force.  Work is a form of energy.  The basic unit of energy in the SI system is joule (J).  One Joule of work is done when 1 Newton moves a particle 1 meter; 1J =  1N•m.

Let’s take a look at the image above.  Particle P is restrained along path S which moves along the x-axis of the defined coordinate system.  There is a certain offset force F that is applied to the particle P.  From this information the energy that force will input into particle P can be defined by the equation below.

Eq 1 $du=Fdscosθ$ or $\int_{s_1}^{s_2}Fcosθds$

Work can also be defined by the area under the force displacement curve as seen in the image below.

# Power

When you know the amount of energy that is required to move a particle you can then calculate the power that was required to move that paticle. Power is an integral of energy in respect to time. Refer to equation 2.

(Eq 2)  $P=F·V=\frac{dU}{dt}$

F = force

V = Velocity

P = Power

dU = Change in Energy

dt = Change in Time

Equation 2 represents the power input for a particle that has a linear projection. To calculate the power of a particle that has a rotational projection equation 3 would be used.

(Eq 3) $P=Tϖ$

T = Torque

ϖ = angular velocity

The units that represent power can be seen in equation 4.

(Eq 4)  $1W=1\frac{J}{S}=1\frac{N·m}{s},~or~1hp = 550\frac{ft·lb}{s}$

Finally, due to energy losses from friction or drag, the power that is put in will be more then the output power. To calculate the efficiency equation 5 would be used.

(Eq 5)$ε=\frac{energy~output}{energy~input}$

Problem

A car is accelerating onto the highway.  It has the capability to go from 0-60 mph in 7 seconds.  The car’s weighs 3500lb.  If the car is accelerating as fast as it can how much energy will be required for the car to reach 60mph?  How much power will need to transmitted to the wheels of the car?  The on-ramp is on a flat surface.

Solution

Step 1:  Determine the cars acceleration.

$60mph = 88\frac{ft}{s}$

$a=\frac{88}{7}=12.6\frac{ft}{s^2}$

Step 2:  Determine the force transmitted to the wheels of the car.

$F=(\frac{3500}{32.2})(12.6)=1370 lb$

Step 3:  Determine the distance the car traveled before reaching full acceleration.

$s=12.6(7^2)=617~ft$

Step 4: Determine the energy required for the car to accelerate to 60mph in 4 seconds.

$U_{1-2}=1370(617)=845290~ft·lb$

Step 5: Determine the power that will need to be transmitted to the wheels of the car.

$P=\frac{845290}{7}=120755\frac{ft·lb}{s}=220~hp$

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