(a) With what speed must a ball be thrown vertically from ground level to rise to a maximum heig

(a) With what speed must a ball be thrown vertically from...

Key Concepts

Graphical Analysis of Motion

Graphing displacement, velocity, and acceleration over time provides a visual representation of the object's motion. Typically, the displacement graph is a parabola, the velocity graph is a straight line with a constant slope equal to the acceleration, and the acceleration graph is a horizontal line reflecting the constant acceleration due to gravity. Marking specific events, such as the moment when maximum height is reached, enhances the understanding of these relationships.

Uniformly Accelerated Motion

This concept refers to motion in which an object experiences a constant acceleration, enabling the use of standard kinematic equations. In vertical motion under gravity, the acceleration remains steady (typically -9.8 m/s² downward), which simplifies the analysis of the object's displacement, velocity, and time of flight.

Kinematic Equations

These equations relate displacement, initial velocity, final velocity, acceleration, and time. They are essential for solving problems involving objects moving under constant acceleration, such as determining the initial speed needed to reach a specified height or the total time an object remains in the air.

Maximum Height in Projectile Motion

In vertical projectile motion, the maximum height is reached when the instantaneous velocity becomes zero. This condition allows for the calculation of the required initial speed using kinematic relationships, since the final velocity at the peak of the trajectory is zero.

Time of Flight

This refers to the total duration an object spends in the air during its motion. For vertical projectile motion, the time to rise to the maximum height equals the time to descend back to the launch level, which allows the total time to be computed as twice the ascent time.

Transcript

00:01 In this example, we're working with a little bit of one -dimensional kinematics.

00:04 So we have motion in one dimension.

00:06 Okay, and what we want to answer is, with what speed must a ball be thrown vertically from the ground to rise to a maximum height of 50 meters? okay, so height max equals 50 meters.

00:21 What does our initial velocity need to be to reach this height? okay, i'm going to use the expression v final squared equals v initial squared plus two times acceleration times distance.

00:39 Okay, i'm going to use this equation coupled with the fact that i know when my ball, so say we launch our ball straight up with some initial velocity.

00:49 Okay, when my ball reaches h max, my velocity is momentarily v, we'll call this vf equals zero before we turn around and come back down.

01:02 Note we're talking only one dimension here.

01:04 I'm only considering vertical motion.

01:05 We don't care about what's going on in the horizontal.

01:08 Okay, so my vertical velocity at this point is zero.

01:11 Okay, so now this equation becomes zero equals v initial squared plus two ad.

01:20 What's our distance now? that distance is height max.

01:23 What's my acceleration? that's the acceleration due to gravity.

01:27 And that's g equals negative 9 .81 meters per second squared in the y direction.

01:34 So this becomes plus negative two gh max.

01:39 Okay, or v initial equals the square root of two gh max.

01:48 Remember, h max is 50 meters.

01:51 Okay, so this gives me an initial velocity of, that's going to be 100 times 9 .81.

01:58 And then we take the square root.

02:00 So an initial velocity of 31 .3 meters per second.

02:06 1 .3 meters per second.

02:10 And note that's in the positive y direction.

02:14 Okay, the next question we want to ask is how long will the ball be in the air? okay, and for that, i'm going to use the equation v final equals v initial plus acceleration times time.

02:29 Again, our final velocity is zero.

02:33 So this becomes v initial equals gt time equals v initial over g equals 3 .19 seconds.

02:50 So not very long to reach that.

02:56 Okay, and okay, so that's the time that it takes for the ball to reach its maximum height...

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