Question

What horizontal distance forward will the paintball have traveled once it reaches that maximum height?

A ramen loving orange ninja builds a paintball catapult with no regard for any safety or protocol. It can lob a paintball into the air at 43 m/s with a slightly less than ideal angle of 50 degrees above the horizontal. What is the maximum height that the ball will reach (above the launch height)? What horizontal distance forward will the paintball have traveled once it reaches that maximum height?
(Warning: You could do a quadratic... or you could do the short way!) Is the paintball moving in the split second it is at the maximum height? Explain why if you happened to use zero in your equations above for velocity.

Answer 1

The maximum height reached by the paintball can be calculated using the vertical component of velocity and the horizontal distance traveled at this point is obtained by calculating the horizontal velocity and the time it takes to reach the maximum height.

Step-by-step explanation:

To find the maximum height reached by the paintball, we will use the vertical component of the initial velocity and the acceleration due to gravity. The initial vertical velocity vy can be calculated by vy = v * sin(θ), where v is the launch speed and θ is the launch angle. The formula used is vy^2 = 2 * g * h, where g is the acceleration due to gravity and h is the maximum height. Plugging in the values vy = 43 m/s * sin(50°) and g = 9.81 m/s^2, we'll get the maximum height after some calculation.

To find the horizontal distance the paintball traveled when it reached the maximum height, we need the time t to reach that height and the initial horizontal velocity vx. The horizontal distance is then x = vx * t, where vx = v * cos(θ). Since at maximum height the vertical velocity is zero, we can use t = vy / g from the motion in vertical direction to find the time to reach maximum height.

The paintball is still moving horizontally at the maximum height as the horizontal motion is independent of the vertical motion in projectile motion, explaining why horizontal velocity is not zero at the peak height.