A body is projected downward at an angle of 30.0^∘ with the horizontal from the top of a buildin

step 1 Today we're looking at a simple example of projectile motion. All right, so say you're standing

A body is projected downward at an angle of 30.0^∘ with the...

Key Concepts

Projectile Motion

Projectile motion refers to the two-dimensional motion of an object that is launched into the air and moves under the influence of gravity. This concept is crucial as it involves analyzing the horizontal and vertical motions separately, assuming that the only force acting on the object (after launch) is gravity, which constantly accelerates the object downward.

Decomposition of Initial Velocity

Decomposing the initial velocity involves splitting the velocity vector into its horizontal and vertical components using trigonometric functions such as sine and cosine. This process is fundamental for solving projectile motion problems because it allows the independent analysis of the motion in each direction.

Kinematic Equations

Kinematic equations relate displacement, velocity, acceleration, and time for objects undergoing constant acceleration, such as in projectile motion where gravity is the only acceleration acting on the object. These equations are essential tools for calculating important parameters like time of flight, displacement, and final velocities.

Quadratic Equations

In the context of projectile motion, solving for time often results in a quadratic equation, especially when determining the time at which the projectile reaches a particular vertical position. Understanding how to solve quadratic equations is key to finding the correct time of flight and ensuring accurate calculations in the analysis.

Horizontal Range

The horizontal range of a projectile is the total horizontal distance it covers from the initial launch point to where it lands. It is calculated by multiplying the horizontal component of the velocity by the time of flight. This concept helps in determining where the projectile will strike relative to its launch point.

Resultant/Impact Angle

The impact angle is the angle at which the projectile strikes the ground relative to the horizontal. It can be derived by computing the arctan of the ratio of the vertical velocity component to the horizontal velocity component at the moment of impact, reflecting how gravity has influenced the vertical velocity during the flight.

Transcript

00:02 Today we're looking at a simple example of projectile motion.

00:07 All right, so say you're standing on top of a building.

00:11 There's some windows.

00:15 All right, and we have a ball and we want to throw it down at the ground and we're throwing it down at an angle of 30 degrees with respect to the horizontal.

00:25 So that's my theta, theta equals 30.

00:29 And we're throwing it with an initial velocity of 40, meters per second.

00:35 So that's my b not equals 40 meters per second.

00:45 And the height of our building here is 170 meters.

00:52 Right.

00:52 And we want to find out a couple things about this system.

00:54 We want to find the time it's going to take for the ball to reach the ground.

00:58 So we want time.

00:59 We want to know how far in the x direction the ball is going to impact with the so i'll call that my dx.

01:07 And we want to find the impact angle here with respect to the horizontal data.

01:12 So i will just call that data final.

01:15 All right, we want to find these three quantities with the information that the problem gives us.

01:22 All right.

01:23 So to find the time, we want to use the motion in the y direction because we know there's no forces acting horizontally.

01:29 We have a constant velocity.

01:31 That's not going to tell us anything about time.

01:33 But in the y direction, we have the acceleration.

01:35 Due to gravity, we can use that to find time.

01:40 So the equation i'm going to use is delta y equals b0y times time plus 1 half a t squared.

01:51 All right, and we know from our projectile motions equations that v0 y equals v .0 sine of theta.

02:02 All right, since we're moving in the downward direction, this is negative.

02:09 And then our acceleration is due to gravity, also in the downward direction.

02:14 So that's negative 1 .5 gt squared.

02:19 All right.

02:23 So let me put in some numbers here.

02:26 So we have negative 170 equals negative 40 sign of 30 minus 1 half g is 4 .9.

02:39 4 .9...

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