Acceleration: $a = \frac{(27.8 \text{ m/s}) - (0 \text{ m/s})}{10 \text{ s}} = 2.78 \text{ m/s}^2$
Force: $F = (1000 \text{ kg})(2.78 \text{ m/s}^2) = 2780 \text{ N}$
Distance: $d_s = \frac{a \cdot dt^2}{2} = \frac{(2.78 \text{ m/s}^2)(10 \text{ s})^2}{2} = 139 \text{ m}$
Work: $W = (2780 \text{ N})(139 \text{ m}) = 386420 \text{ J}$
Power: $P = \frac{386420 \text{ J}}{10 \text{ s}} = 38642 \text{ W} = 38.6 \text{ kW}$

Imperial Units:

Acceleration: $a = \frac{(91.1 \text{ ft/s}) - (0 \text{ ft/s})}{10 \text{ s}} = 9.11 \text{ ft/s}^2$
Force (using slug conversion): $F = \left(\frac{2205 \text{ lbm}}{32.17405 \text{ lbm/slug}}\right)(9.11 \text{ ft/s}^2) = 624 \text{ lbf}$
Distance: $d_s = \frac{(9.11 \text{ ft/s}^2)(10 \text{ s})^2}{2} = 455 \text{ ft}$
Work: $W = (624 \text{ lbf})(455 \text{ ft}) = 284075 \text{ ft·lbf}$
Power: $P = \frac{284075 \text{ ft·lbf}}{10 \text{ s}} = 28407 \text{ ft·lbf/s}$

Note: Real-world car acceleration varies due to gear shifts and motor characteristics.

Benchmark Standards

The following standard benchmark velocities are commonly used to evaluate and compare car acceleration performance:

0 to 60 mph: Equivalent to 0 to 26.8 m/s or 0 to 96.6 km/h.
0 to 100 km/h: Equivalent to 0 to 27.8 m/s or 0 to 62.1 mph.

Important Physical Properties

Average vs. Instantaneous Acceleration: The formulas provided calculate average acceleration over a given time interval or distance. Real-world acceleration varies continuously due to gear shifts, engine torque curves, and traction limits.
Forces in Real-World Scenarios: The calculated acceleration force (F = m a) represents the net force required assuming no other forces. In practice, the engine must also overcome aerodynamic drag and rolling friction, which reduce the net force available for acceleration.
Work-Energy Relationship: The work done on the car (W = F l) equals the change in its kinetic energy during the acceleration phase.
Power Requirement: The power (P = W / dt) indicates the rate at which work is done. Higher acceleration (shorter dt) requires significantly more power.

Key Unit Conversions

The following conversion factors are essential for calculations across metric and Imperial systems:

1 ft·lbf = 1.35582 J
1 ft·lbf/s = 1.35582 W
1 hp = 550 ft·lbf/s = 745.7 W

Additional Notes

Imperial Mass Unit Conversion

In the Imperial system, mass is measured in slugs rather than pounds-mass (lbm):

$1 \text{ slug} = 32.17405 \text{ lbm}$

To convert lbm to slugs for force calculations:

$m_{\text{slugs}} = \frac{m_{\text{lbm}}}{32.17405}$

Imperial Energy and Power Conversions

When working in Imperial units, the following conversions apply:

$1 \text{ ft·lbf} = 1.36 \text{ J}$

$1 \text{ ft·lbf/s} = 1.36 \text{ W} = 0.00182 \text{ hp}$

Calculation Assumptions

Important: The force, work, and power formulas presented here calculate values for mass acceleration only. Forces due to air resistance (drag) and rolling friction are not included in these calculations.

In real-world conditions, actual vehicle acceleration performance will differ from these idealized calculations due to:

Aerodynamic drag — increases with the square of velocity
Rolling resistance — tire and drivetrain friction losses
Gear shifts — discrete ratio changes affect acceleration profile
Motor characteristics — torque and power curves vary with RPM

Imperial Units and Conversions

When working in the Imperial system, mass is typically measured in slugs, not pounds-mass (lbm). The conversion factor is fundamental for calculating forces correctly.

1 slug = 32.17405 lbm
Force (lbf) = Mass (lbm) * Acceleration (ft/s²) / 32.17405

Additional Formulas

Acceleration from Distance and Time

If the distance moved and the time are known instead of velocities, the constant acceleration can be calculated as:

a = \frac{2 ds}{dt^2}

where ds is the distance moved (m, ft).

Acceleration Work

The work done (W) by the acceleration force over a distance is:

W = F \cdot l

where F is the acceleration force (N, lbf) and l is the distance moved (m, ft).

Acceleration Power

The average power (P) required for acceleration is the work done divided by the time interval:

P = \frac{W}{dt}

where W is work (J, ft·lbf) and dt is time (s).

Real-World Considerations

The theoretical calculations assume constant acceleration. In reality, the acceleration of a vehicle is not constant and varies due to:

Gear Shifts: Changes in transmission gear ratios alter the torque and force at the wheels.
Motor Characteristics: Engine torque output varies with RPM, affecting acceleration throughout the speed range.

References

Original Source

Source: engineeringtoolbox.com