A. A projectile is launched with speed v0 and angle θ....

a. A projectile is launched with speed $v_{0}$ and angle $\theta .$ Derive an expression for the projectile's maximum height $h$ b. A baseball is hit with a speed of $33.6 \mathrm{m} / \mathrm{s}$. Calculate its height and the distance traveled if it is hit at angles of $30.0^{\circ}$ $45.0^{\circ},$ and $60.0^{\circ}$

a. A projectile is launched with speed $v_{0}$ and angle $\theta .$ Derive an expression for the projectile's maximum height $h$
b. A baseball is hit with a speed of $33.6 \mathrm{m} / \mathrm{s}$. Calculate its height and the distance traveled if it is hit at angles of $30.0^{\circ}$ $45.0^{\circ},$ and $60.0^{\circ}$

Key Concepts

Projectile Motion

Projectile motion involves the analysis of an object launched into the air where its only acceleration is due to gravity. This motion is characterized by a parabolic trajectory, and key parameters such as launch speed, launch angle, and gravitational acceleration determine its path, maximum height, and range.

Kinematic Equations

Kinematic equations describe the motion of objects under uniform acceleration. They are used to derive expressions for various quantities, including displacement, velocity, and time. In projectile motion, these equations help derive the maximum height reached and the total horizontal distance traveled.

Decomposition of Initial Velocity

The initial launch velocity of a projectile can be split into vertical and horizontal components using trigonometric functions. The vertical component (v0*sin?) is crucial in determining the maximum height reached, as this is the component that is opposed by gravitational acceleration.

Maximum Height Calculation

The maximum height of a projectile is achieved when its vertical velocity component reduces to zero due to gravity. Using kinematic equations, setting the final vertical velocity to zero allows one to derive an expression for maximum height in terms of the initial vertical velocity and acceleration due to gravity.

Projectile Range

The range of a projectile refers to the total horizontal distance covered before it returns to its initial vertical level. It is derived using the horizontal component of the initial velocity and the time of flight, which is influenced by the vertical motion under gravitational acceleration.

Transcript

00:01 For number 41, we're given that projectiles launched at an angle of theta, and v -initial v .0 will be the velocity.

00:08 And we're to find an expression for the height.

00:11 So i know that for the vertical motion, i know that acceleration be negative g, the initial velocity will be this component.

00:26 So that's the opposite side.

00:28 So i'll use sine.

00:30 So sine of theta times v.

00:36 The, it's going to go up until it stops, so v final will be zero if i'm doing just the way up, and i'm looking for the vertical displacement.

00:48 So i'm going to use the f squared equation.

01:03 Vf would be zero, the i will be this, that gets squared, so i'm going to put that squared and that squared to negative g times d, and i'm solving for this d.

01:21 So i'm solving for this d.

01:22 So so imagine to take this all to this side, so it's positive.

01:26 You get the d by itself.

01:35 So i bring the 2 and the g over here.

01:41 So this would be the vertical displacement or the height.

01:44 So there's my expression.

01:46 So that's my answer to part a.

01:56 For part b, i'm going to know i'm given the velocity of a baseball is 33 .6 meters per second.

02:07 And then i'm to find how high and how far it goes when it's thrown at 30 degrees, 45 degrees, and 60 degrees.

02:18 So i have my expression for height.

02:21 I could do that for all of them.

02:23 Maybe i'll go ahead and do that.