A small rock is thrown vertically upward with a speed of 22.0 m/s from the edge of the roof of a 30.0-m-tall building. The rock doesn't hit the building on its way back down and lands on the street below. Ignore air resistance. (b) How much time elapses from when the rock is thrown until it hits the street?

Verified step by step guidance

1. Step 1: Determine the initial velocity of the rock, which is given as 22.0 m/s upward. Let this be denoted as $v_0 = 22.0 \, \text{m/s}$.
2. Step 2: Calculate the total displacement from the point of release to the street. Since the rock is thrown from the top of a 30.0 m building, and it lands on the street below, the total displacement $y$ is $-30.0 \, \text{m}$ (negative because the final position is below the initial position).
3. Step 3: Use the kinematic equation for uniformly accelerated motion that relates initial velocity, acceleration, displacement, and time: $y = v_0 t + \frac{1}{2} a t^2$. Here, $a = -9.8 \, \text{m/s}^2$ (acceleration due to gravity, directed downward).
4. Step 4: Substitute the known values into the equation and solve for $t$. The equation becomes $-30.0 = 22.0 t - 4.9 t^2$. This is a quadratic equation in the form $at^2 + bt + c = 0$.
5. Step 5: Solve the quadratic equation using the quadratic formula $t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. Choose the positive root for $t$ since time cannot be negative.

Key Concepts

Kinematics

Kinematics is the branch of physics that describes the motion of objects without considering the forces that cause the motion. It involves concepts such as displacement, velocity, acceleration, and time. In this problem, kinematic equations will be used to analyze the rock's vertical motion, allowing us to calculate the time it takes to reach the ground.

Acceleration due to Gravity

Acceleration due to gravity is the rate at which an object accelerates towards the Earth when in free fall, typically denoted as 'g' and approximately equal to 9.81 m/s². This constant affects the rock's motion after it is thrown, as it will decelerate while moving upward and accelerate downward after reaching its peak height. Understanding this concept is crucial for determining the total time of flight.

Projectile Motion

Projectile motion refers to the motion of an object that is thrown into the air and is subject to gravitational forces. It can be analyzed in two dimensions, but in this case, we focus on the vertical component. The rock's initial upward velocity and the height of the building will influence its trajectory and the time it takes to hit the ground, making this concept essential for solving the problem.