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The speed of the wheels determines the speed of the engine, while the torque produced by the engine determines the acceleration of the wheels.
Let:
[[$ S_v $]] be the speed of the vehicle in [[$m/s$]]
[[$ m $]] be the mass of the vehicle in [[$ kg $]]
[[$ A_v $]] be the acceleration of the vehicle in [[$ m/s^2 $]]
[[$ S_\omega $]] be the speed of the engine in [[$ rad / s $]]
[[$ S_e $]] be the speed of the engine in [[$ rpm $]]
[[$ T_e $]] be the torque produced by the engine in [[$ Nm $]]
[[$ R_d $]] be the gear ratio of the driveline
[[$ E_d $]] be the efficiency of the driveline
[[$ r_w $]] be the radius of the wheels in metres
[[math]]
S_\omega = \frac {S_v} {R_d r_w}
[[/math]]
[[math]]
A_v = \frac {T_e E_d} {R_d r_w m}
[[/math]]
Since [[$ rpm = \frac {rad}{s} \frac {60}{2\pi} $]], then [[$ S_e = S_\omega \frac {60}{2\pi} $]], and so we get
[[math]]
S_e = \frac {30 S_v} {R_d r_w \pi}
[[/math]]
or:
[[math]]
S_e = 9.5493 \frac {S_v } {R_d r_w}
[[/math]]
Since [[$ S = \int A \, dt $]],
[[math]]
S_v = \int_0^t A_v \, dt = \int_0^t \frac {T_e E_d} {R_d r_w m} \, dt
[[/math]]
[[math]]
S_\omega = \frac { \int_0^t \frac {T_e E_d} {R_d r_w m} \, dt}{R_d r_w}
[[/math]]
[[math]]
S_e = 9.5493 \frac { \int_0^t \frac {T_e E_d} {R_d r_w m} \, dt}{R_d r_w}
[[/math]]
[[math]]
S_e = 9.5493 \, \frac {E_d} {R_d^2 \, r_w^2 \, m} \int_0^t T_e \, dt
[[/math]]
The speed of the wheels determines the speed of the engine, while the torque produced by the engine determines the acceleration of the wheels.
Let:
$S_v$ be the speed of the vehicle in $m/s$
$m$ be the mass of the vehicle in $kg$
$A_v$ be the acceleration of the vehicle in $m/s^2$
$S_\omega$ be the speed of the engine in $rad / s$
$S_e$ be the speed of the engine in $rpm$
$T_e$ be the torque produced by the engine in $Nm$
$R_d$ be the gear ratio of the driveline
$E_d$ be the efficiency of the driveline
$r_w$ be the radius of the wheels in metres
(1)
\begin{align} S_\omega = \frac {S_v} {R_d r_w} \end{align}
(2)
\begin{align} A_v = \frac {T_e E_d} {R_d r_w m} \end{align}
Since $rpm = \frac {rad}{s} \frac {60}{2\pi}$, then $S_e = S_\omega \frac {60}{2\pi}$, and so we get
(3)
\begin{align} S_e = \frac {30 S_v} {R_d r_w \pi} \end{align}
or:
(4)
\begin{align} S_e = 9.5493 \frac {S_v } {R_d r_w} \end{align}
Since $S = \int A \, dt$,
(5)
\begin{align} S_v = \int_0^t A_v \, dt = \int_0^t \frac {T_e E_d} {R_d r_w m} \, dt \end{align}
(6)
\begin{align} S_\omega = \frac { \int_0^t \frac {T_e E_d} {R_d r_w m} \, dt}{R_d r_w} \end{align}
(7)
\begin{align} S_e = 9.5493 \frac { \int_0^t \frac {T_e E_d} {R_d r_w m} \, dt}{R_d r_w} \end{align}
(8)
\begin{align} S_e = 9.5493 \, \frac {E_d} {R_d^2 \, r_w^2 \, m} \int_0^t T_e \, dt \end{align}
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