A home run is hit in such a way that the baseball just...

A home run is hit in such a way that the baseball just clears a wall $21 \mathrm{~m}$ high, loeated $130 \mathrm{~m}$ from home plate. The ball is hit at an angle of $35^{\circ}$ to the horizontal, and air resistance is negligible. Find (a) the initial speed of the ball, (b) the time it takes the ball to reach the wall, and (c) the velocity components and the speed of the ball when it reaches the wall. (Assume the ball is hit at a height of $1.0 \mathrm{~m}$ above the ground.)

Key Concepts

Solving Simultaneous Equations

In problems involving projectile motion, simultaneous equations are often used to relate the horizontal and vertical motions. For example, determining the initial speed or time of flight may require solving equations that incorporate both kinematic relationships. This method is crucial when specific conditions, like obstacle clearances or designated landing points, must be satisfied.

Kinematic Equations

Kinematic equations provide the mathematical relationships between displacement, initial velocity, time, and acceleration. In projectile motion, these equations are applied separately to horizontal (with zero acceleration) and vertical motion (with constant acceleration due to gravity) to determine unknown quantities such as time to reach a certain point, velocity components at a given instant, and the overall range.

Trajectory Analysis

Trajectory analysis involves understanding the parabolic path of a projectile. By using the initial speed and launch angle, one can derive the trajectory equation that relates the vertical position to the horizontal position. This analysis is critical in determining points of interest along the path, such as the peak height or the location where the projectile clears an obstacle.

Projectile Motion

Projectile motion refers to the two?dimensional motion of an object under the influence of gravity, where the only acceleration acting on the object is due to gravity. This concept is essential for analyzing the motion of any object, such as a baseball, when air resistance is negligible, and it helps predict the path, time of flight, and other properties of the motion.

Decomposition of Motion into Components

This concept involves breaking down the object’s initial velocity into horizontal and vertical components. Because horizontal and vertical motions are independent (with constant horizontal velocity and uniformly accelerated vertical motion), analyzing them separately simplifies problem solving and allows for the application of kinematic equations in each direction.

Transcript

00:01 All right, we've got a home run being hit, and it just clears a wall that is 21 meters high.

00:09 So it's going over 130 meters, and the height in the end is 21 meters.

00:21 All right.

00:22 It's hit at 35 degrees, and it starts, h0 is 1 meter.

00:42 35 degrees.

00:47 Okay.

00:52 A, what's the initial speed of the ball? okay.

00:57 Well, h is going to equal h initial plus v sine theta t minus, minus 1⁄2g t squared.

01:25 Also, d is going to equal 0 plus v cosine theta t.

01:37 So, t is d over v cosine theta.

01:50 So i'll substitute that into the first equation, h equals h0 plus v0 sine theta times, why do i write zero on that? d over v cosine theta minus one -half -g d over v cosine theta squared.

02:25 Okay, now we'll simplify this.

02:28 H minus h -0.

02:36 Those vs cancel out minus d tangent theta.

02:45 It's going to be negative one -half -g d.

02:49 Squared over v squared, cosine squared, theta.

02:54 All right.

02:56 So if i multiply by 2 over g, i get 2 over g.

03:03 I mean multiply by negative 2 over g.

03:07 H0 plus d tangent theta minus h is going to equal d squared over v squared, cosine squared of theta.

03:21 And so, v is going to be d over cosine theta square root of g over 2, h0, plus d tangent theta minus h.

03:44 Okay, putting that in a calculator, i know theta is 35.

03:57 I know that h0 is 1.