A boy tosses a ball onto the roof of a house. For the launch...

A boy tosses a ball onto the roof of a house. For the launch conditions shown, determine the slant distance $s$ to the point of impact. Also, determine the angle $\theta$ which the velocity of the ball makes with the roof at the moment of impact.

Key Concepts

Projectile Motion

Projectile motion describes the path of an object under the influence of only gravity after an initial thrust. It involves analyzing the horizontal and vertical components separately, accounting for constant horizontal velocity and uniformly accelerated vertical motion.

Kinematics Equations

Kinematics equations are mathematical formulas that relate displacement, velocity, acceleration, and time in motion problems. They are essential for solving projectile motion problems by providing the tools necessary to compute distances, speeds, and impact timings.

Vector Decomposition

Vector decomposition is the process of breaking down an object’s velocity or force into perpendicular components. This is crucial in projectile motion, as it allows for separate analysis of the horizontal and vertical motions, and in determining the angle of impact relative to an inclined surface.

Inclined Plane Analysis

Inclined plane analysis involves understanding how angles and slopes affect motion. In projectile motion problems involving a sloped surface, it includes calculating the effective distance along the incline (slant distance) and the angle between the object's velocity vector and the plane at the point of impact.

Trigonometry in Physics

Trigonometry in physics is used to relate angles to side lengths in geometric configurations. In the context of projectile motion, it facilitates the determination of trajectory angles, the projection of vectors, and the calculation of distances and impact angles on inclined surfaces.

Transcript

00:01 I've drawn a diagram and we're just asked to find s and theta.

00:05 So let's find the angle before we can do anything else.

00:09 So our angle of the roof, so the tangent will be 5 over 12, and that will be 22 .62 degrees.

00:32 So the sign of theta is 5 over 13, and the cosine, or this alpha, this is my.

00:42 Angle here that's my alpha is 12 over 13 sign was 5 over 13 okay the coordinates of the rock that he's throwing a rock as a function of time so coordinates of rock as a function of time and let's do an x will be and then putting the first one to the second one let's put first equation into the second equation and we can position the position of the rock when it hits the roof x will be equal to and y will be equal to and then let's put our values in so we're going to have four plus s times 5 over 13 equals 1 .75 plus 9 plus s 12 over 13 times the tangent of 50 and this will be minus 9 .81 times 2 times 12 .5 squared times the cosine squared of 50 times 81 plus 18 s 12 over 3 13 plus s squared 12 over 13 squared.

03:11 My goodness.

03:12 So then we solve for this.

03:13 We're going to get 0 .064 -738 s squared.

03:20 Plus 0 .546928s minus 2 .321619 equals 0.

03:32 And our positive choice for s will be 3 .104 meters.

03:44 So 3 .104 meters.

03:49 That's our answer for s.

03:51 Now to find our theta, let's find the time that this occurs.

04:06 And we'll use our equations for our x.

04:23 So solving for t here, we'll get 1 .4767, because we know what that is.

04:32 And we know what that is.

04:39 Okay, and our angle then, so i'm going to get the tangent of this value will equal, it'll equal gt...