A football player is kicked with an initial velocity u at an angle to the horizontal and reaches the ground t seconds later. Ignoring air resistance, explain why the range R of the football is ut(costheta)
The range R of the football can be calculated using the formula R = ut(costheta), where u is the initial velocity, t is the time taken to reach the ground, and costheta is the cosine of the launch angle.
1. Horizontal Motion: When the football is launched at an angle, it can be broken down into horizontal and vertical components. The horizontal motion is unaffected by gravity. This means that the football maintains a constant horizontal velocity throughout its flight.
2. Time of Flight: The time taken for the football to reach the ground is denoted by t. This time is the same for both horizontal and vertical motions because gravity affects both components equally.
3. Horizontal Velocity: The horizontal component of velocity (Vx) remains constant during the entire flight of the football. Therefore, the distance traveled horizontally, which is the range R, can be calculated by multiplying the horizontal velocity by the time taken, as R = Vx * t.
4. Horizontal Velocity in Terms of Initial Velocity and Launch Angle: The horizontal velocity (Vx) can be expressed as Vx = u * costheta, where u is the initial velocity and costheta is the cosine of the launch angle. This is because the horizontal component of the initial velocity is u * costheta.
By substituting the expression for Vx into the range formula, we get R = (u * costheta) * t, which simplifies to R = ut(costheta). This equation represents the range of the football, ignoring air resistance.