Question

A rock is tossed off the top of a cliff of height \(34.9 \mathrm{~m}\). Its initial speed is \(29.3 \mathrm{~m} / \mathrm{s}\), and the launch angle is \(29.9^{\circ}\) above the horizontal. What is the speed with which the rock hits the ground at the bottom of the cliff?

Short answer

Answer: To find the final speed of the rock, first calculate the initial components of velocity v_0x and v_0y, then find the time it takes for the rock to reach the ground using the vertical motion equation. Next, determine the final horizontal and vertical components of velocity, and finally, use the Pythagorean theorem to calculate the final speed of the rock.

Step-by-step solution

Find the Initial Components of Velocity

Given initial speed, we can find the initial components of the velocity: \(v_{0x}\) and \(v_{0y}\). \(v_{0x} = v_0 \cos{\theta} = 29.3 \cos{29.9^\circ}\), \(v_{0y} = v_0 \sin{\theta} = 29.3 \sin{29.9^\circ}\)

Calculate the Time it Takes for the Rock to Reach the Ground

We can use the equation for vertical motion to find the time: \(y_f = y_0 + v_{0y}t - \frac{1}{2}gt^2\). Since the rock is being tossed from the top of the cliff, \(y_f =0\) and \(y_0 =34.9m\). We have the initial vertical velocity \(v_{0y}\) from step 1, and the acceleration due to gravity \(g=9.81ms^{-2}\). Solving for the time \(t\): \(0 = 34.9 + v_{0y}t - \frac{1}{2}(9.81)t^2\)

Find the Final Components of Velocity

With the time \(t\), we can find the final horizontal and vertical components of velocity. For horizontal velocity, there is no acceleration, so the final horizontal velocity is the same as the initial horizontal velocity: \(v_{fx} = v_{0x}\) For vertical velocity, we use the equation \(v_{fy} = v_{0y} - gt\). \(v_{fy} = v_{0y} - (9.81)t\)

Calculate the Final Speed of the Rock

Now we have both the final horizontal and vertical components of velocity, we can use the Pythagorean theorem to determine the final speed of the rock: \(v_f = \sqrt{v_{fx}^2 + v_{fy}^2}\)

Key concepts

Kinematics

Kinematics is the branch of mechanics that deals with the motion of objects, without reference to the forces that cause such motion. It provides a framework for understanding how objects move through space and time. Let's consider the rock being tossed off a cliff. In kinematics, we describe its motion using variables such as displacement, velocity, and acceleration. The projectile motion of the rock can be analyzed by breaking it into horizontal and vertical components. Since the rock's horizontal movement is unaffected by gravity, it will maintain its initial horizontal velocity throughout the journey. Conversely, the vertical motion is influenced by gravitational acceleration, which only affects the vertical component of the rock's velocity, causing it to change over time. Practically, this approach allows us to use equations of kinematics separately in the horizontal and vertical directions to determine the positional and velocity changes of the rock until it hits the ground.

For students wrestling with kinematics, it's important to visualize the motion, perhaps by sketching the trajectory of the projectile. This will help in understanding how the two components of motion combine to define the rock's path.

Velocity Components

In projectile motion, velocity components are a crucial concept that enable us to analyze the motion in a more manageable way. Every velocity vector can be split into horizontal and vertical components. For our cliff-tossed rock, the initial velocity, given an angle, is divided using trigonometry: the horizontal component, \(v_{0x}\), and the vertical component, \(v_{0y}\). Understanding these components is key because they act independently of each other - while gravity does not affect \(v_{0x}\), it continuously changes \(v_{0y}\) due to acceleration or deceleration.

When studying projectile motion, you should always first resolve the initial velocity into its components. This step simplifies further calculations like determining flight duration, peak height, and final velocity upon impact. Remembering that horizontal velocity remains constant while vertical velocity changes linearly with time due to gravity will help you master the concepts of velocity components.

Equations of Motion

Equations of motion are the formulas that allow us to connect kinematic quantities and solve for unknowns within a given physical situation. For an object in projectile motion, such as our rock, there are different equations for the horizontal and vertical components of its motion. Horizontally, where acceleration is zero, the position changes linearly over time with constant velocity. Vertically, we use equations that factor in the initial velocity and the constant acceleration due to gravity.

For instance, the vertical motion of the rock can be analyzed with the equation \(y_f = y_0 + v_{0y}t - \frac{1}{2}gt^2\), which factors in initial position, initial vertical velocity, time, and gravitational acceleration. This equation is invaluable for calculating how long it takes the rock to hit the ground. Equations of motion provide a critical toolkit for solving a wide array of physics problems, and practicing their application in various scenarios is essential for grasping their use.

Angular Projection

Angular projection, or the launch angle of a projectile, is a fundamental parameter that influences the trajectory of an object in motion. When the rock is thrown from the top of the cliff with a given angle above the horizontal, this angle determines how the initial speed is partitioned between the horizontal and vertical components of velocity. In the case of our cliff-rock exercise, a launch angle of \(29.9^\circ\) means the rock's initial velocity vector is tilted, resulting in both horizontal and vertical motion upon release.

A projectile launched at an angle will follow a curved path, and the analysis of its motion requires an understanding of how the horizontal and vertical components interact over time. The larger the launch angle, the higher and shorter the path of the projectile. Conversely, a smaller launch angle leads to a longer but less elevated trajectory. Grasping the concept of angular projection is vital to predict and calculate the range, maximum height, and final impact velocity of a projectile.