Question

A projectile is launched at an angle of \(45.0^{\circ}\) above the horizontal. What is the ratio of its horizontal range to its maximum height? How does the answer change if the initial speed of the projectile is doubled?

Short answer

Answer: The ratio of the horizontal range to the maximum height for a projectile launched at an angle of 45 degrees is 4:1. When the initial speed is doubled, the ratio remains unchanged at 4:1.

Step-by-step solution

We need to calculate the initial horizontal (\(v_{0x}\)) and vertical (\(v_{0y}\)) velocities to find the range and maximum height. We can do this using trigonometric functions for the given angle: \(v_{0x} = v_0 \cos(45^{\circ})\) \(v_{0y} = v_0 \sin(45^{\circ})\) Since the angle is \(45^{\circ}\), both \(\cos(45^{\circ})\) and \(\sin(45^{\circ})\) are equal to \(\frac{\sqrt{2}}{2}\). Thus, we can write: \(v_{0x} = \frac{v_0\sqrt{2}}{2}\) \(v_{0y} = \frac{v_0\sqrt{2}}{2}\) #Step 2: Derive the expressions for the horizontal range and maximum height#

Using the equations of motion and the initial horizontal and vertical velocities, we can find expressions for the horizontal range (R) and maximum height (H) as follows: 1. Horizontal Range (R): \(R = \frac{v_{0x} \times T}{\cos(45^{\circ})}\), where T is the total flight time of the projectile. 2. Maximum height (H): \(H = \frac{v_{0y}^2}{2g}\), where g is the acceleration due to gravity. By substituting \(v_{0x}\) and \(v_{0y}\) from step 1 and rearranging, we arrive at the expressions needed: \(R=\frac{v_0^2 \sin(2 \times 45^{\circ})}{g}\) \(H=\frac{v_0^2 \sin^2(45^{\circ})}{2g}\) #Step 3: Calculate the ratio of horizontal range to maximum height as R:H#

To find the ratio, we will divide the expression for R by the expression for H: $$\frac{R}{H} = \frac{\frac{v_0^2 \sin(2 \times 45^{\circ})}{g}}{\frac{v_0^2 \sin^2(45^{\circ})}{2g}} = \frac{2 \sin(2 \times 45^{\circ})}{\sin^2(45^{\circ})}$$ Since \(\sin(90^{\circ})=1\) and \(\sin^2(45^{\circ})=\frac{1}{2}\), we have: $$\frac{R}{H} = \frac{2 \times 1}{\frac{1}{2}} = 4$$ Thus, the ratio of horizontal range to maximum height is 4:1. #Step 4: Determine the new ratio when initial speed is doubled#

If the initial speed is doubled, we will update our expressions for R and H: \(R=\frac{(2v_0)^2 \sin(2 \times 45^{\circ})}{g}\) \(H=\frac{(2v_0)^2 \sin^2(45^{\circ})}{2g}\) Now we compute the new ratio: $$\frac{R}{H} = \frac{\frac{(2v_0)^2 \sin(2 \times 45^{\circ})}{g}}{\frac{(2v_0)^2 \sin^2(45^{\circ})}{2g}} = \frac{2 \sin(2 \times 45^{\circ})}{\sin^2(45^{\circ})}$$ Since the values of \(\sin\) do not change, the new ratio remains the same and equals 4:1. Thus, doubling the initial speed does not change the ratio of horizontal range to maximum height for a projectile launched at an angle of \(45^{\circ}\).

Key concepts

Horizontal Range

The horizontal range of a projectile refers to the distance it covers in the horizontal direction before touching the ground again. It represents one of the two key measurements in projectile motion, with the other being the maximum height reached. To find the horizontal range, we use the formula:

\[ R = \frac{v_{0}^2 \sin(2\theta)}{g} \]

Where \(v_{0}\) is the initial velocity, \(\theta\) is the angle of projection, and \(g\) stands for the acceleration due to gravity. A fascinating aspect of the horizontal range is that it is directly influenced by both the launch angle and the initial speed of the projectile. The greatest range is achieved with a launch angle of \(45^\circ\), which gives the ideal balance of vertical and horizontal motions. Doubling the initial velocity while keeping the angle constant will quadruple the range, highlighting the quadratic relationship between initial speed and range.

Maximum Height

The maximum height of a projectile is the peak vertical position it attains during its flight. This is a crucial concept in understanding projectile motion since it reflects the highest altitude the projectile reaches above its launch point. The equation to determine the maximum height is:

\[ H = \frac{v_{0y}^2}{2g} \]

In this equation, \(v_{0y}\) is the initial velocity in the vertical direction, which can be derived from the initial speed and the angle of projection using trigonometric functions. Despite the intuitive belief, increasing the initial speed of the projectile does not affect the ratio of the horizontal range to maximum height when the launch angle is \(45^\circ\), due to the symmetrical properties of the projectile's parabolic trajectory at this angle. Understanding this concept helps in making predictions about the flight of various projectiles, from sports balls to rockets.

Trigonometric Functions in Physics

Trigonometric functions like sine and cosine are invaluable tools in physics, especially when dealing with projectile motion. They enable us to decompose the initial velocity into horizontal and vertical components, essential for analyzing the motion. For instance, with a launch angle \(\theta\):

\[ v_{0x} = v_0 \cos(\theta) \]\[ v_{0y} = v_0 \sin(\theta) \]

These components are crucial in obtaining the horizontal range and maximum height of the projectile as they are plugged into the respective formulas. In the scenario of a projectile launched at an angle of \(45^\circ\), the cosine and sine functions both yield \(\frac{\sqrt{2}}{2}\), granting equal parts of velocity to horizontal and vertical motions. This symmetry at \(45^\circ\) is special because it maximizes the range for a given initial velocity. In more complex problems, other trigonometric functions such as tangent and their inverses might be used to determine additional characteristics of projectile motion.

Equations of Motion

The equations of motion are fundamental to physics; they describe the relationship between an object's various kinematic quantities, including displacement, velocity, acceleration, and time. When analyzing projectile motion, the following two key equations are often employed:

\[ R = \frac{v_{0x} \times T}{\cos(\theta)} \] \[ H = \frac{v_{0y}^2}{2g} \]

The horizontal range equation considers the horizontal velocity and the time of flight, while the maximum height equation involves the square of the vertical velocity and the constant gravitational acceleration. It's interesting to note that the ratio of the horizontal range to the maximum height is preserved under certain conditions of symmetry, such as a launch angle of \(45^\circ\), regardless of changes in the initial speed. A thorough understanding of these equations is essential for students to predict and analyze the motion of projectiles under various circumstances.