Consider a projectile that is launched from the origin of an...

Consider a projectile that is launched from the origin of an $x y$ coordinate system with speed $v_{i}$ at initial angle $\theta_{i}$ above the horizontal. Note that at the apex of its trajectory the projectile is moving horizontally, so that the slope of its path is zero. Use the expression for the trajectory given in Equation $4.12$ to find the $x$ coordinate that corresponds to the maximum height. Use this $x$ coordinate and the symmetry of the trajectory to determine the horizontal range of the projectile.

Key Concepts

Projectile Motion

Projectile motion describes the motion of an object that is thrown or projected into the air and is subject only to the acceleration of gravity. The object’s path is typically a parabola, and its motion can be analyzed separately in the horizontal and vertical directions using kinematic equations, under the assumption of negligible air resistance.

Trajectory Equation

The trajectory equation is the mathematical representation of the projectile’s path, usually expressed as a quadratic function relating the vertical displacement to the horizontal displacement. This equation is derived by combining the independent horizontal and vertical equations of motion, and it captures the parabolic nature of the path.

Maximum Height

The maximum height is the peak point in the projectile’s trajectory where its vertical component of velocity becomes zero. At this point, the projectile’s motion is entirely horizontal. Analyzing the maximum height involves setting the derivative of the vertical position with respect to horizontal displacement to zero and solving for the corresponding coordinate.

Symmetry in Projectile Motion

Symmetry in projectile motion refers to the fact that the path of a projectile is symmetric about the mid-point of its horizontal range. This means that the time and horizontal distances taken by the projectile to ascend to its maximum height are equal to those taken to descend from the maximum height to the ground, which is used to calculate the total horizontal range.

Horizontal Range

The horizontal range of a projectile is the total horizontal distance traveled from the launch point to the point of impact with the ground. Determining the range involves using the trajectory equation, particularly by finding the horizontal positions corresponding to the launch and landing points, and making use of the symmetry of the flight path.

Transcript

0:00 Hello.

00:02 In this problem, we have a projectile that is launched from the origin with angle theta with the horizon.

00:11 So this is how the projectile looks like.

00:15 And there is a hint in this problem that says that at the apex of this motion, so this is the apex, which is the maximum height that a projectile can reach.

00:32 At this point, a particle is only moving in the...

00:35 Horizontal direction and his vertical velocity is zero.

00:39 So basically this means that the slope of the trajectory path here is zero.

00:45 So at the apex, slope of the trajectory is zero.

00:54 The hint now is using equation 4 .12 that states the following.

01:01 This is y and it's equal to 10 of theta initial multiplied by x minus g divided by two v i squared cosine squared theta initial multiplied by x squared this is a relation that relates both y and x so this is the equation of the trajectory and it's not a surprise that this is a formula for a parabola which is uh contains the power of two in terms of x.

01:42 So using the fact that the slope of the trajectory is zero at this point, we're asked to give an expression for the x using this equation.

01:51 So add the slope.

01:53 So let's first get the slope.

01:55 The slope is basically d .y by dx.

02:00 And let's take the derivative of this equation respect to x.

02:04 Here, the only part that will be left is 10 theta initial...

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