Final Answer:
Tripling the initial velocity of the pumpkin while keeping the launch angle fixed and less than 90 degrees will result in a ninefold increase in its range.
Step-by-step explanation:
When analyzing projectile motion, the range of a projectile depends on its initial velocity, launch angle, and gravitational acceleration. The formula for the range (R) in projectile motion is given by:
R = v_0^2 .sin(2θ}/{g}
where \(v_0\) is the initial velocity, \(\theta\) is the launch angle, and \(g\) is the acceleration due to gravity. In this scenario, the launch angle is fixed, and we are examining the effect of tripling the initial velocity (\(v_0 \rightarrow 3v_0\)). Plugging this into the range formula, we get:
R' ={(3v_0)^2 .sin(2θ}/{g}
Simplifying this expression yields:
R' = 9 . {v_0^2 .sin(2θ}/{g}
Therefore, the range with tripled initial velocity (R') is nine times the original range (R). This demonstrates a direct proportionality between the square of the initial velocity and the range, provided the launch angle remains constant. The ninefold increase in range is a consequence of the squared term in the velocity component of the formula.
In practical terms, this means that if the pumpkin's initial velocity is tripled, it will cover a significantly greater horizontal distance before hitting the ground, making the projectile travel much farther while maintaining the same launch angle. This phenomenon is crucial in understanding how changes in initial conditions impact the trajectory and reach of projectiles in projectile motion scenarios.