Schre's Leap: Projectile Motion of a Cat Treated as a Point Mass

Problem Statement

One day, a physics student observed his cat "Schre" make a remarkable jump.

Schre jumped from a horizontal floor, barely cleared the tip of a potted plant (an obstacle) placed in front, and landed perfectly on a favorite toy on the floor.

The student, using the physicist's bold assumption that "a cat is a point mass," set out to analyze the initial velocity of this extraordinary jump.

Define the launch point as the origin $O(0, 0)$O(0,0), with the $x$x-axis pointing horizontally to the right and the $y$y-axis pointing vertically upward.

Schre is launched from the origin at elevation angle $\theta$θ ($0 < \theta < \frac{\pi}{2}$0<θ<2π​) and initial speed $v_0$v0​. The tip of the obstacle is at coordinates $(L, H)$(L,H), and the toy is at $(D, 0)$(D,0).

Air resistance is negligible; the magnitude of gravitational acceleration is $g$g.

From the conditions that Schre's trajectory passes through both $(L, H)$(L,H) and $(D, 0)$(D,0), derive the value of $v_0^2$v02​.

Constraints

• Horizontal distance to obstacle: $L = 1.0 \text{ m}$L=1.0 m
• Height of obstacle: $H = 0.50 \text{ m}$H=0.50 m
• Horizontal distance to landing point: $D = 1.6 \text{ m}$D=1.6 m
• Magnitude of gravitational acceleration: $g = 9.8 \text{ m/s}^2$g=9.8 m/s2

Input Format

Compute $v_0^2$v02​ using the given values. The answer is an irreducible fraction $\frac{A}{B}$BA​ ($A, B$A,B are coprime positive integers); give the value of $A + B$A+B as a positive integer.