Which equation describes the relationship between horizontal velocity and range? This is a fundamental question in physics, particularly in the study of projectile motion. Understanding this relationship is crucial for various applications, such as calculating the distance a projectile will travel and determining the optimal angle for launching a projectile to achieve the maximum range. In this article, we will delve into the equation that governs this relationship and explore its implications in different scenarios.
The equation that describes the relationship between horizontal velocity (v_x) and range (R) is given by:
R = v_x t
where t represents the time of flight. This equation assumes that the projectile is launched at an angle θ with respect to the horizontal and that there is no air resistance. The range of the projectile is the horizontal distance it travels before hitting the ground.
To derive this equation, we can break down the projectile’s motion into horizontal and vertical components. The horizontal component of the velocity (v_x) remains constant throughout the flight, while the vertical component (v_y) changes due to the acceleration due to gravity (g).
The horizontal distance traveled by the projectile (R) is the product of its horizontal velocity (v_x) and the time of flight (t). The time of flight can be determined by the vertical motion of the projectile. Since the projectile is launched at an angle θ, the vertical component of the initial velocity (v_yo) is given by:
v_yo = v sin(θ)
where v is the initial velocity of the projectile. The vertical displacement (h) of the projectile is given by:
h = v_yo t – (1/2) g t^2
Since the projectile returns to the same height from which it was launched, the total vertical displacement is zero. Therefore, we can set h equal to zero and solve for t:
0 = v_yo t – (1/2) g t^2
Solving for t, we get:
t = (2 v sin(θ)) / g
Substituting this expression for t into the equation for range (R), we obtain:
R = v_x t
R = v cos(θ) (2 v sin(θ)) / g
R = (v^2 sin(2θ)) / g
This equation shows that the range of a projectile is directly proportional to the square of its initial velocity and the sine of twice the launch angle, and inversely proportional to the acceleration due to gravity.
Understanding this relationship can help us optimize the launch angle and velocity for maximum range. For example, when launching a projectile at a 45-degree angle, the range is maximized. This is because sin(2 45°) = 1, which makes the term (v^2 sin(2θ)) / g equal to its maximum value.
In conclusion, the equation that describes the relationship between horizontal velocity and range is a powerful tool in the study of projectile motion. By understanding this equation, we can predict the distance a projectile will travel and optimize its launch parameters for various applications.
