The development of Adaptive Cruise Control (ACC) systems aims to enhance the safety and comfort of vehicles by automatically regulating the speed of the vehicle to ensure a safe gap from the preceding vehicle. However, conventional ACC systems are unable to adapt themselves to changing driving conditions and drivers' behavior. To address this limitation, we propose a Long Short-Term Memory (LSTM) based ACC system that can learn from past driving experiences and adapt and predict new situations in real time. The model is constructed based on the real-world highD dataset, acquired from German highways with the assistance of camera-equipped drones. We evaluated the ACC system under aggressive lane changes when the side lane preceding vehicle cut off, forcing the targeted driver to reduce speed. To this end, the proposed system was assessed on a simulated driving environment and compared with a feedforward Artificial Neural Network (ANN) model and Model Predictive Control (MPC) model. The results show that the LSTM-based system is 19.25% more accurate than the ANN model and 5.9% more accurate than the MPC model in terms of predicting future values of subject vehicle acceleration. The simulation is done in Matlab/Simulink environment.