A ball is thrown horizontally from the top of a 60.0- m building and lands 100.0 m from the bas

A ball is thrown horizontally from the top of a 60.0- m...

A ball is thrown horizontally from the top of a 60.0- $\mathrm{m}$ building and lands $100.0 \mathrm{~m}$ from the base of the building. Ignore air resistance. (a) How long is the ball in the air? (b) What must have been the initial horizontal component of the velocity? (c) What is the vertical component of the velocity just before the ball hits the ground? (d) What is the velocity (including both the horizontal and vertical components) of the ball just before it hits the ground?

Key Concepts

Projectile Motion

Projectile motion describes the motion of an object that is thrown or projected into the air, subject only to the acceleration due to gravity. It is a two-dimensional motion where the horizontal and vertical components are independent, allowing us to analyze each separately. This concept is fundamental in solving problems where an object is launched with an initial velocity and follows a parabolic path due to gravity.

Kinematics Equations

Kinematics equations are the mathematical tools used to describe an object's motion under constant acceleration. These equations relate displacement, initial velocity, final velocity, time, and acceleration. They are essential for determining quantities such as time of flight, horizontal displacement, and the final velocity components in projectile motion problems.

Decomposition of Motion

Decomposition of motion involves breaking down a projectile's motion into horizontal and vertical components. Since there is no acceleration in the horizontal direction (assuming no air resistance) and constant acceleration in the vertical direction due to gravity, each component can be analyzed separately. This approach simplifies the analysis of projectile trajectories and is key to solving related problems.

Time of Flight Calculation

Calculating the time of flight in projectile motion typically involves analyzing the vertical motion under gravity. By applying the appropriate kinematics equations to the vertical component, one can determine how long the projectile remains in the air before reaching the ground. This time is the same for both horizontal and vertical analyses in projectile motion problems.

Velocity Components Analysis

Analysis of velocity components involves determining both the horizontal and vertical components of the velocity at any given point during the projectile's motion. The horizontal component remains constant throughout the motion (in the absence of air resistance), while the vertical component changes linearly with time due to gravity. Understanding these components is crucial for determining the overall velocity vector and its magnitude at impact.

Transcript

00:01 So first, how long is the ball in the air? we have y final equals y initial plus v y initial t plus one half gt squared.

00:10 Now, we know that here we're going to say that y initial is zero meters.

00:17 If y initial is zero meters, that means that y final is 60 meters.

00:22 But if we're choosing downwards to be positive, that means that the acceleration due to gravity has to to be positive as well.

00:32 And so we know that this ball is being thrown perfectly horizontally initially.

00:38 So the vy initial is going to be zero.

00:41 So this term becomes zero.

00:43 And then we can solve for t.

00:45 So t is then going to be equalling the square root of two times y final divided by g.

00:53 And again, if we're choosing downwards to be positive, which is completely okay because we can choose our frame of reference, this would be equaling the square root of 2 times 60 meters divided by 9 .8 meters per second squared.

01:09 And we find that the time in the air 3 .5 seconds, this would be our answer for part a.

01:16 Now, for part b, what must have been the initial component of the velocity? well, this is very easy in the sense that we can say that x final is equaling vx initial times t.

01:30 And we know that there isn't any acceleration in the x direction, so we don't have to account for the acceleration term.

01:37 And so we can say that vx initial is simply equalling x final, which we know is 100 meters, because the ball lands 100 meters away from the base of the building, divided by the time in the air 3 .5 seconds we found in part a...

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