Answer
The initial speed of the sports car was 21 m/s.
Work Step by Step
We can use $\mu_k$ to find the magnitude of deceleration after the collision.
$ma = mg~\mu_k$
$a = (9.80~m/s^2)(0.80)$
$a = 7.84~m/s^2$
We can use the acceleration to find the speed $v$ of both cars as they started to slide together.
$v^2 = 2ax$
$v = \sqrt{2ax} = \sqrt{(2)(7.84~m/s^2)(2.6~m)}$
$v = 6.38~m/s$
We can use conservation of momentum to find the initial speed $v_1$ of the sports car.
$m_1~v_1 = (m_1+m_2)~v$
$v_1 = \frac{(m_1+m_2)(v)}{m_1}$
$v_1 = \frac{(980~kg+2300~kg)(6.38~m/s)}{980~kg}$
$v_1 = 21~m/s$
The initial speed of the sports car was 21 m/s.
