To calculate the distance, velocity, and gravity involved in these situations, we can use the equations of motion. For the archer, we can use the equation d = v*t, where d is the distance, v is the velocity, and t is the time. In this case, we are given the distance (42.0 m) and velocity (95.0 m/s), so we can solve for the time it takes for the arrow to reach the target. The equation becomes t = d/v, so t = 42.0 m / 95.0 m/s = 0.442 seconds.
Now, we can use the equation d = 1/2 * g * t^2, where g is the acceleration due to gravity (9.8 m/s^2) and t is the time. We know the time (0.442 seconds) and we want to find the distance above the bull's-eye, so we rearrange the equation to solve for d, giving us d = 1/2 * g * t^2 = 1/2 * 9.8 m/s^2 * (0.442 s)^2 = 0.975 m. This is the distance the arrow will drop due to gravity, so the archer should aim 0.975 m above the bull's-eye to compensate for this drop.
For the beach ball scenario, we can use the same equations. We are given the velocity (1.30 m/s) and the distance traveled (0.73 m), so we can solve for the time using the equation t = d/v. This gives us t = 0.73 m / 1.30 m/s = 0.562 seconds.
Next, we can use the equation d = 1/2 * g * t^2 to find the height of the pier. We know the time (0.562 seconds) and we want to find the height, so we rearrange the equation to solve for h, giving us h = d / (1/2 * g * t^2) = 0.73 m / (1/2 * 9.8 m/s^2 * (0.562 s)^2) = 0.73 m / 2.76 = 0.264 m. This is the height of the pier above the water.
In summary, by using the equations of motion, we can calculate the distance
