Power needed to accelerate up incline

[ferswin]

Homework Statement

a vehicle of mass 200kg accelerates from rest up to 40km/h in 15seconds up an incline of 1 in 8. The resistance to motion can be considered to be in two parts, the first being due to friction and constant at 50 N/tonne with the second part being due to air resistance and being equal to 0.02v2 + v Newtons, where v is in km/hr.
a) Calculate the resistance to motion at 40km/h
b) Calculate the effort required to achieve 40km/h
c) Calculate the power required

d) we are later asked as part of coursework to choose a electric motor size for vehicle

Homework Equations

The Attempt at a Solution

a)whilst traveling at constant velocity

Force of air resistance (N) = 0.02*40^2+40 = 72 N
Rolling resistance (N) = 50 * 0.2t = 10 N
Force of incline (N) = 200*9.81*1/8 = 245.25

total resistance to motion at 40km/h (N) = 72+10+245.25 (N) = 327.25 N

b) Effort required = total resistance to motion + ma

to find a

v = 40km/h = 11.1m/s
u = 0 m/s
t = 15 s
a = ? m/s^2

v = u + at
11.1 (m/s)= a (m/s^2)*15(s)
a = 0.74 m/s^2

effort required = E = 327.25 (N)+ (200(kg)*0.74(m/s^2))
E = 475.25 N

c)
to find work done we need distance travelled

s = ut + 0.5at^2
s = 0.5*0.74(m/s^2)*15^2(t)
s = 83.25 m

workdone = 475.25(N)*83.25(m) = 39564.6 joules

Power = workdone/time = 39865.6(j)/15(s) = 2637.6 j/s (watts)

Power (hp) = 2598.2 / 745.7 = 3.49 hp

the answer I obtained doesn't take into account the varing air resistance would i need to find an average and recalculate?

in terms of selecting a suitable size motor if i work out the power needed at 39.9km/h with a acceleration of 0.74m/s^2 the power needed is

power = (total resistance + ma)*v

power = (475.25(N)+0.74(m/s^2)*200(kg))*11.1(m/s)
power = 6918 j/s

power (hp) = 6918/754.7 = 9.28 hp

I'm a little confused as why the power required to reach 40km/h is so less than power needed at 39.9km/h with a acceleration 0.74m/s^s. have i done calc wrong? what size motor do i go for? please help.

 

[Hootenanny]

Your solution to (a) is correct. However, for (b) is the resistance to motion going to be constant throughout the acceleration?

 

[ferswin]

hi, thanks, do I need to try and find an average for air resistance?

 

[Hootenanny]

hi, thanks, do I need to try and find an average for air resistance?

The effort required is going to be a function of velocity. Tell me have you solved any first order differential equations before? If not, I may be sending you off on a wild goose chase.

 

[ferswin]

no haven't done any 'first order differential equations before' will this be hard? the main of the cousework was to select a suitable motor and battery supply.

 

[Hootenanny]

no haven't done any 'first order differential equations before' will this be hard? the main of the cousework was to select a suitable motor and battery supply.

What's your level? Have you done any calculus previously?

 

[ferswin]

it is BEng degree level (UK), I have gone back into education after many years so I'm quite rusty. but i know we haven't covered any differentiating in this course yet.

 

[Hootenanny]

I think I've over-complicating things a little here anyway. What you've done for (b) looks okay to me. However, for (c) I believe that you're going to have to take into account the varying air resistance as you say. Note that for motion in one dimension we define the work done by a variable force thus;

W = \int^{x_1}_{x_0}F\cdot dx

 

[ferswin]

thanks, i'll some research into this, see if i can sovle it now