# Power needed to accelerate up incline

## 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

## The Attempt at a Solution

a)whilst travelling 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.

## The Attempt at a Solution

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

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

Hootenanny
Staff Emeritus
Gold Member
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.

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
Staff Emeritus
Gold Member
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?

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