### Auto Motion

#### Power Requirements for Constant Speed and Constant Acceleration

For constant speed v, the net force must be zero:
F = f_{air} + f_{rolling}

Power required to maintain speed v is: P = Fv

Alternatively: W = F x and P = W/t = F (x/t) = F v.

For v = 30 m/s (≈ 65 mph),

f_{air} ≈ 500 N ,
f_{rolling} ≈ 300 N → F ≈ 800 N.

Then P = F v = 800 x 30 = 24,000 W ≈ 32 hp !!

To move at constant acceleration a:
F - f_{air} - f_{rolling} = ma

Typical numbers: 0 - 30m/s (≈ 65 mph) in 10 sec

→ a = 3
m/s^{2}, and m= 1500 kg

Then F = ma + f_{air} +
f_{rolling} = 4500 + 300 + ( < 500)

→ max. force ≈ 5300
N.

Max. power (at top speed):

P_{max}
= F_{max} v_{max} = 5300 x 30 = 160 KW ≈ 210 hp
!!

#### Power and Acceleration

A car goes from 0 to 30 mph in 1.5 seconds. How long does it take the car to
from from 0 to 60 mph?
*Assume that the power of the engine is velocity
independent. Neglect friction and wind resistance.*

- 3 sec
- 4.5 sec
- 6 sec
- 9 sec
- 12 sec

Definition of power: P= dE/dt = d (½ mv^{2})/dt = constant.

This implies v^{2} ∝ t, or v ∝
t^{1/2}.

Acceleration is decreasing in time!