Final answer:
To find the distance a cart will travel before coming to rest when starting at a different speed on an air track, kinematic equations and energy conservation are used. Assuming constant deceleration, the stopping distance when the initial speed is doubled will be four times greater, due to the quadrupling of initial kinetic energy.
Step-by-step explanation:
The student question revolves around a scenario where a cart on an air track slows down and stops after the air is turned off. To find out how far a cart will travel before coming to rest when it starts at a different speed, we can use the principles of kinematics and energy conservation.
In the first scenario, the initial speed (vi) is 0.5 m/s, and it travels 1 m before stopping. Assuming deceleration is constant, we can calculate the deceleration using the kinematic equation vf2 = vi2 + 2ad, where vf is the final speed (0 m/s), a is the acceleration (deceleration in this case), and d is the distance (1 m).
When the speed doubles (to 1 m/s), we can predict the stopping distance under the assumption that the deceleration remains constant using the same kinematic equation, solving for the new distance. By plugging in the values, we would find that the cart travels four times the distance (4 m) before coming to a stop due to the quadrupling of the initial kinetic energy that needs to be dissipated by the work done against friction.