A rock is thrown vertically upward with a speed of 12.0 m / s from the roof of a building that i

A rock is thrown vertically upward with a speed of 12.0 m /...

A rock is thrown vertically upward with a speed of 12.0 $\mathrm{m} / \mathrm{s}$ from the roof of a building that is 60.0 $\mathrm{m}$ above the ground. (a) In how many seconds after being thrown does the rock strike the ground? (b) What is the speed of the rock just before it strikes the ground? Assume free fall.

Key Concepts

Kinematics

Kinematics is the branch of mechanics that involves the study of motion without considering the forces that cause the motion. It uses equations that relate displacement, velocity, acceleration, and time to analyze trajectories and motion paths, especially in cases of constant acceleration.

Projectile Motion

Projectile motion refers to the motion of an object thrown or projected into the air, subject to only the acceleration due to gravity. This concept involves analyzing the vertical and horizontal components separately, although in vertical motion problems the analysis focuses primarily on the motion under gravity.

Free Fall

Free fall describes the motion of objects under the sole influence of gravity, with negligible air resistance. In free fall, an object experiences a constant acceleration, typically represented by the acceleration due to gravity, allowing for the use of standard kinematic equations to predict its motion.

Quadratic Equations in Kinematics

In kinematics, solving problems related to the time of flight or displacement frequently leads to quadratic equations. These equations arise from the integration of acceleration into velocity and displacement, and the quadratic formula is often used to obtain the correct time intervals or position values.

Transcript

00:01 So we will take positive y to be upward and ground to be on the zero level.

00:13 This means that final way, final distance from ground is zero because it reaches the ground.

00:26 And it starts from some initial height.

00:31 So why not is not zero? so let's say it starts from here, which corresponds to y0, and this point corresponds to y, and it falls downward.

00:41 So, y0 minus y is 60 meter.

00:50 Now to find the time, let's use this equation.

01:03 Now, since the ball is, sorry, since the rock is dropped.

01:11 From rest so b0 y is zero and acceleration is acceleration due to gravity but it points downward so this is equal to negative 9 .8 meter per second square so y minus y not i think i made a mistake over here yeah so initial velocity doesn't start the rock doesn't start from rest it's dropped with some initial velocity and that is given to be 12 meter per second so we use these values in this equation so we have y minus y not to be negative of y0 minus y so this is negative 60 meter which is equal to velocity initial velocity times t plus half of negative 9 .8 which is negative 4 .9 meter per second square times t square so this is a quadratic equation and we solve it so this will give us two values so those are 12 plus minus square of negative 12 minus 4 times 4 point nine times negative 60 and if we simplify this we get time to be 1 .22 plus minus 3 .7 1 seconds so if you take the negative sign your time will come out to be negative so that's not physical so that's not physically possible so the only time will be for the positive value.

03:38 So that should be equal to 1 .2 plus 3 .71 which is 4 .93 seconds.

03:49 Now if you are wondering about how to arrive at this equation, so let me write a general formula.

03:59 So let's say this is your quadratic equation and x is your t here.

04:08 So x will be given by i need some more space...

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