Cooperative adaptive cruise control entails regulating the distance towards a preceding road vehicle, employing both on-board environmental sensors and wireless inter-vehicle communication, the latter necessary for acceleration information, which cannot be directly measured by on-board sensors. However, in case of communication impairments, such as high latency or significant packet loss, a degraded mode is required, to which end the acceleration of the preceding vehicle must be estimated using on-board sensors such as radar. Fast convergence towards the actual preceding vehicle acceleration is imperative for ensuring the cooperative driving safety. Consequently, we propose a sliding-mode based estimator ensuring the finite-time convergence of the acceleration estimation error, and hence, ensuring the functionality of the CACC in degraded situations.