A ball is thrown straight up from the edge of the roof of a...

A ball is thrown straight up from the edge of the roof of a building. A second ball is dropped from the roof 1.00 s later. You may ignore air resistance. (a) If the height of the building is $20.0 \mathrm{m},$ what must the initial speed of the first ball be if both are to hit the ground at the same time? On the same graph, sketch the position of each ball as a function of time, measured from when the first ball is thrown. Consider the same situation, but now let the initial speed $v_{0}$ of the first ball be given and treat the height $h$ of the building as an unknown. (b) What must the height of the building be for both balls to reach the ground at the same time (i) if $v_{0}$ is 6.0 $\mathrm{m} / \mathrm{s}$ and (ii) if $v_{0}$ is 9.5 $\mathrm{m} / \mathrm{s} ?(\mathrm{c})$ If $v_{0}$ is greater than some value $v_{\max }$ a value of $h$ does not exist that allows both balls to hit the ground at the same time. Solve for $v_{\text { max. The value }} v_{\text { max }}$ has a simple physical interpretation. What is it? (d) If $v_{0}$ is less than some value $v_{\text { min }}$ a value of $h$ does not exist that allows both balls to hit the ground at the same time. Solve for $v_{\text { main }}$ The value $v_{\text { min }}$ also has a simple physical interpretation. What is it?

Key Concepts

Graphical Representation of Motion

Drawing graphs of position versus time helps visualize the motion of objects under constant acceleration. These graphs can illustrate the effects of differences in initial velocity and time delays, and they provide insight into the dynamics of the system by showing intersections (such as simultaneous impacts) and trends in the motion.

Physical Interpretation of Critical Speeds

In some problems, constraints lead to limits on the initial speed that allow a particular outcome. The concepts of maximum and minimum initial speeds arise from the physical requirements of matching trajectories or timings, and they often correspond to natural physical quantities, such as the free-fall speed from a specific height or speed needed to overcome gravitational delay.

Projectile Motion in One Dimension

This concept focuses on objects moving under the influence of gravity in a single vertical direction. It involves analyzing the upward and downward parts of the trajectory separately while using the same kinematic equations, making it possible to calculate parameters like maximum height, time of flight, and impact velocity.

Uniformly Accelerated Motion

This concept involves the study of objects under a constant acceleration, such as gravity acting on a body in free fall. The motion is described using kinematic equations that relate displacement, initial velocity, time, and constant acceleration, allowing one to predict the future position and velocity of the object.

Kinematic Equations

These are the mathematical expressions that relate an object’s displacement, initial velocity, time, and acceleration in motion. They are essential for solving problems involving projectile motion, free fall, or any situation where acceleration is constant, and they allow for predicting various outcomes such as time of flight and impact conditions.

Time Delay and Relative Motion

Time delay in motion refers to the analysis of objects that begin their motion at different instants. Understanding how to incorporate the difference between start times into the kinematic equations is crucial for problems where two bodies with different initial conditions need to be synchronized at a particular event, such as hitting the ground simultaneously.

Transcript

00:01 Now this question we have two balls that are being thrown off the edge.

00:04 So the first ball is actually thrown upwards.

00:07 Right, at a certain velocity.

00:10 So we're going to call this v.

00:14 And then the other ball is just released at one second later.

00:20 So at t equals to one second.

00:23 Then this ball, the second ball is released from the roof without throwing.

00:31 Now to find the initial speed in which you will both of these balls will actually hit the ground at the same time.

00:40 You want to find what is this given that the height of this building is 20 meters.

00:56 So the convention, you can choose any direction to be positive or negative.

01:04 It's up to you but i'm going to choose.

01:07 Upwards to be positive and i'm gonna take upwards positive means that the position right from the original position is negative 20 right when you reach the ground so you're gonna use the equation be sorry s equals to u t plus half a t squared for the first ball, the one that gets thrown up, it gets thrown up with an upward velocity, so that's a positive velocity.

01:55 And it travels for a certain time t.

01:59 Acceleration is due to gravity, so that's negative.

02:03 Negative 9 .1, which i'm going to just leave it as g for now, times t squared.

02:09 And the displacement is negative 20.

02:17 So just not to confuse you, this is v times t.

02:24 Now for the other case we have the other ball that is dropped directly downwards.

02:31 Displacement is also 20 but the initial velocity is zero.

02:35 So it has no initial velocity.

02:37 It's just half times negative half times gt prime square.

02:43 So t prime is slightly different.

02:46 T prime is the total time required for it to fall...