A golf ball is struck at ground level. The speed of the golf...

A golf ball is struck at ground level. The speed of the golf ball as a function of the time is shown in Fig. 4-27, where $t=0$ at the instant the ball is struck. The scaling on the vertical axis is set by $v_{a}=19 \mathrm{~m} / \mathrm{s}$ and $v_{b}=31 \mathrm{~m} / \mathrm{s}$. (a) How far does the golf ball travel horizontally before returning to ground level? (b) What is the maximum height above ground level attained by the ball?

Key Concepts

Projectile Motion

Projectile motion describes the motion of an object that is launched into the air and is subject only to the force of gravity (ignoring air resistance). This concept involves analyzing two independent components: a horizontal component with constant velocity and a vertical component with constant acceleration due to gravity. The method used to solve such problems relies on separating these components and applying the appropriate kinematic equations to determine quantities like range, time of flight, and maximum height.

Horizontal and Vertical Components

The initial velocity of a projectile is typically resolved into horizontal and vertical components. The horizontal component remains constant throughout the motion since there is no acceleration in that direction (ignoring air resistance), which simplifies the calculation of horizontal displacement. The vertical component is influenced by gravity, causing changes in vertical speed; this leads to the determination of the time until the object returns to its original vertical position and consequently helps in calculating maximum height.

Kinematic Equations

Kinematic equations govern the relationships between displacement, velocity, acceleration, and time. In projectile motion problems, these equations are used separately for the horizontal (constant velocity) and vertical (constant acceleration) directions. For instance, the vertical displacement and final velocity equations are used to calculate the point at which the vertical component of the velocity becomes zero (at maximum height) and when the projectile returns to ground level.

Maximum Height and Range

The maximum height of a projectile is reached when the vertical velocity component becomes zero. By finding the time to reach this point using vertical kinematics, one can calculate the peak altitude using the appropriate displacement equation. Similarly, the range (horizontal distance traveled) is calculated by determining the total time the projectile is in the air and then multiplying it by the constant horizontal speed. These concepts are essential in solving problems involving the trajectory of an object launched from ground level.

Transcript

00:01 So in this problem we have a golf ball being hit.

00:04 We want to find how far and how high it goes.

00:07 So we know this is going to take some parabolic path.

00:09 We're going to have an initial velocity v0.

00:15 We can define it as an angle, but in this case we'll see we can use the components.

00:20 Instead it makes it a bit easier.

00:22 Let's pick our origin to be where the ball is hit, x and y by convention.

00:29 So we can read the plot of velocity which looks something like this.

00:37 31 meters per second at the top, 19 meters per second is a slow speed from zero seconds to five seconds.

00:49 So what this tells us at zero seconds our speed is 31 meters per second, so v0, and speed is the magnitude of velocity.

00:57 So we don't know which direction it is yet, but we know the absolute magnitude.

01:03 Now the slowest speed is 19 meters per second and that's going to correspond to the middle of the flight and the apex of the flight because there's some y velocity upwards that adds to the total magnitude of velocity before and then some y velocity after that would add to it.

01:22 So there's no y velocity right in the middle.

01:24 Now x velocity is constant because there's no acceleration in that direction, so the minimum velocity ends up being right at the top.

01:31 It's horizontal and 19 meters per second.

01:35 And we know this whole thing takes five seconds to reach back to the same point because without air resistance the energy that we spend going up we're going to get coming back down, so we're going to land at the same velocity.

01:50 So our 19 meters per second is our vx which is constant throughout and this directly gives us part a, how far does it travel.

01:59 Distance is velocity times time, so the golf ball traveled 95 meters...

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