Final answer:
By using kinematic equations of motion, we can calculate the time it will take for Hayden to catch up with Matthew after he starts accelerating. The distance each travels must be equal at the point of meeting, accounting for Matthew's constant speed and Hayden's acceleration after a 1.63 second delay.
Step-by-step explanation:
To determine the time it takes for Hayden to catch up with Matthew, we need to use the kinematic equations of motion. Since Matthew is moving at a constant speed, his equation for displacement will be d = v_m \( \cdot \) t_m, where d is the displacement, v_m is Matthew's velocity, and t_m is Matthew's time of travel.
For Hayden, who starts accelerating after a delay of 1.63 seconds, we need to use the equation for displacement considering acceleration: d = 0.5 \( \cdot \) a \( \cdot \) t_h^2, where a is Hayden's acceleration and t_h is Hayden's time of travel from the moment he starts accelerating. To be side by side with Matthew, Hayden must cover the distance Matthew has traveled plus the distance Matthew covers during Hayden's acceleration.
Let d_m be the distance Matthew has traveled during Hayden's 1.63 second delay. Then Matthew has gone d_m = v_m \( \cdot \) 1.63. When Hayden starts accelerating, both will travel the same additional distance to meet at the same point. So, the equation will be:
d_m + v_m \( \cdot \) t_h = 0.5 \( \cdot \) a \( \cdot \) t_h^2. Additionally, we need to subtract 1.63 seconds from Hayden's travel time to account for his initial delay: t_h = t_m - 1.63.
We can solve these equations simultaneously to find t_h, Hayden's travel time from the start of his acceleration until he catches up with Matthew.