The range R of a projectile is defined as the magnitude of...

The range $R$ of a projectile is defined as the magnitude of the horizontal displacement of the projectile when it returns to its original altitude. (In other words, the range is the distance between the launch point and the impact point on flat ground.) A projectile is launched at $t=0$ with initial speed $v_{\mathrm{i}}$ at an angle $\theta$ above the horizontal. (a) Find the time $t$ at which the projectile returns to its original altitude. (b) Show that the range is $$R=\frac{v_{i}^{2} \sin 2 \theta}{g}$$ [Hint: Use a trigonometric identity from Appendix A.7.]

The range $R$ of a projectile is defined as the magnitude of the horizontal displacement of the projectile when it returns to its original altitude. (In other words, the range is the distance between the launch point and the impact point on flat ground.) A projectile is launched at $t=0$ with initial speed $v_{\mathrm{i}}$ at an angle $\theta$ above the horizontal. (a) Find the time $t$ at which the projectile returns to its original altitude. (b) Show that the range is
$$R=\frac{v_{i}^{2} \sin 2 \theta}{g}$$
[Hint: Use a trigonometric identity from Appendix A.7.]

Key Concepts

Kinematic Equations

Kinematic equations describe the motion of objects under constant acceleration. In the context of projectile motion, these equations are used to express the horizontal displacement as a function of time and the vertical displacement as a function of time, which facilitates solving for critical parameters like time of flight and maximum height.

Projectile Motion

Projectile motion is a type of two-dimensional motion in which an object is launched into the air and moves under the influence of gravity alone, neglecting air resistance. Its trajectory is determined by the initial launch conditions and can be analyzed by separating the motion into horizontal and vertical components.

Decomposition of Motion

This concept involves breaking down the projectile's motion into two independent components: horizontal motion with constant velocity and vertical motion with constant acceleration due to gravity. This separation simplifies the analysis and calculation of various quantities such as time of flight and range.

Time of Flight

The time of flight is the total time a projectile remains in the air before returning to its original elevation. It is determined by analyzing the vertical component of the motion using kinematic equations, specifically by setting the vertical displacement to zero and solving for time.

Range of a Projectile

The range of a projectile is the total horizontal distance traveled during its flight. It is computed by multiplying the constant horizontal velocity by the time of flight. The derived expression for range typically involves the launch speed, the acceleration due to gravity, and the sine of twice the launch angle.

Trigonometric Identities

Trigonometric identities, such as the double-angle formula sin(2?) = 2 sin? cos?, are mathematical tools used to simplify expressions involving trigonometric functions. In the derivation of the range formula, this identity allows the expression of the product of sine and cosine of the launch angle in a more compact form, leading to the final formula for the range.

Transcript

00:01 Here we're going to do some equation work.

00:03 So we're told that the range of a projectile is defined as the magnitude of its horizontal displacement.

00:13 In other words, the range is the distance between the launch point and the impact point on flat ground.

00:21 Because we're working with flat ground, there's a couple consequences.

00:24 Let's start by breaking this down.

00:25 So we're saying that a projectile is launched with initial speed v.

00:29 V.

00:29 I, an initial angle above the horizontal theta.

00:33 I'm going to try that right here, where we have a vector of the velocity, again with magnitude v.

00:39 I, with angle above the horizontal theta.

00:43 If we were to break this down into its components, we'd have the x, where v.

00:51 X is equal to the magnitude v.

00:53 I times the cosine of the angle, and then we'd have the v where that is equal to v sub i times the sign of the angle because it's opposite the angle.

01:08 Since we're working with flat ground, the v sub y initial, which is v.

01:12 I times sine data, will be equal in magnitude to v.

01:18 Y final, except they'll be opposite in sign.

01:22 Initially it leaves the ground like this, and when it returns to the ground, this vector and this vector are the exact same in x and y, but the y's are opposite in sign.

01:39 We can back that up with our third equation by saying that v.

01:45 Y final squared is equal to v.

01:48 Y, initial squared, plus 2 times a, times a change in y.

01:53 Well, what is the change in y between the initial and final position? they're at the same height.

01:59 So that change in y is zero, which means this whole term goes to zero.

02:03 In other words, the square of the two is equal, in other words, the magnitude of v.

02:09 F is equal to the magnitude of v...

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