How do I derive equations for Time of Flight and Range in Parabolic Motion?
A ball is thrown with a certain launch angle (theta) above the horizontal with a certain velocity v. Let us assume that the gravitational acceleration g is constant over the flight, the ball has no spin and there are no effects due to air friction. Furthermore, for simplicity, let's say the ball lands at the same height it was thrown and the motion is purely 2-dimensional ( only horizontal and vertical motion ). Velocity is a vector and hence can be decomposed in its horizontal and vertical components, respectively, at the istant when the ball is thrown: v0x = v0*cos(theta) and v0y = v0*sin(theta). Let us calculate first the Time of Flight. By using one of the eqs. for straight motion under constant acceleration: v = v0 + at, we can calculate the time it takes to the ball to reach the top of its parabola ( where vy = 0 ). Hence, 0 = v0y -gt => t = v0sin(theta)/g. By doubling this time to take into account the time it takes to the ball to fall: tflight = 2v0sin(theta)/g. The Range R is defined as the horizontal distance the ball travels over its flight. Note: horizontal velocity can be regarded as constant as there's no force acting on the ball in the horizontal direction ( contrarily to vertical direction where gravitational force acts and provides gravitational acceleration g ). Hence, by eq. x = vt, we can deduce: R = v0x*tflight = v0cos(theta)2v0sin(theta)/g = v02sin(2theta)/g. [ note trig. identity 2cos(theta)sin(theta) = sin(2theta) has been used in last passage ] .
