A plane, diving with constant speed at an angle of 53.0^∘ with the vertical, releases a projecti

step 1 Here we have a projectile motion problem. And first, we want to find the speed of the plane from

A plane, diving with constant speed at an angle of 53.0^∘...

A plane, diving with constant speed at an angle of $53.0^{\circ}$ with the vertical, releases a projectile at an altitude of $730 \mathrm{~m}$. The projectile hits the ground $5.00 \mathrm{~s}$ after release. (a) What is the speed of the plane? (b) How far does the projectile travel horizontally during its flight? What are the (c) horizontal and (d) vertical components of its velocity just before striking the ground?

Key Concepts

Projectile Motion

Projectile motion refers to the motion of an object that is launched into the air and moves under the influence of gravity alone. The object's trajectory is parabolic. Critical parameters such as time of flight, horizontal displacement, and final velocity components are determined using the kinematic equations of motion with gravity acting on the vertical motion.

Kinematics in Two-Dimensions

This concept involves analyzing motion by separating it into horizontal and vertical components. Each component is treated independently: the horizontal component has constant velocity (assuming no air resistance), while the vertical component is influenced by constant acceleration due to gravity. Understanding this separation is key to solving motion problems where an object is launched or released with an initial velocity at an angle.

Velocity Component Decomposition

This is the process of breaking a velocity vector into its horizontal and vertical components using trigonometric functions. When the velocity is given at an angle, knowing how to decompose it is essential to correctly analyze the motion of the projectile, especially when determining its speed, range, and the effects of gravity on its vertical movement.

Initial Conditions in Projectile Motion

Initial conditions include the projectile's initial speed, angle of release, and launch altitude, all of which significantly affect its trajectory. These conditions determine important outcomes such as the time until impact, horizontal displacement, and the magnitudes of the velocity components just prior to landing.

Transcript

00:01 Here we have a projectile motion problem.

00:02 And first, we want to find the speed of the plane from which the projectile is launched.

00:10 So to do that, what we can do is first think about how we can find one of the components of the velocity so that we can use trigonometry to find the velocity.

00:25 We know the time and we know the height.

00:27 So let's try to find the vertical component of velocity first.

00:36 Now we could use the following equation.

00:39 We can say that height is equal to the initial velocity in the y -axis, which is what we are trying to find, times the time, plus one half over multiplied, i'm sorry, by gravity times times squared.

01:00 Now notice that we're writing plus acceleration because we're defining down to be positive in this case.

01:08 So now when we try to solve for the initial velocity in the y -axis, we obtain that it is equal to the height divided by the time minus one -half times the gravity times the time.

01:31 And we have all the information to complete this equation.

01:33 We know that the height is 730 meters.

01:37 We know that the time is 5 seconds minus 1 half times gravity times the time and with that we obtain that the velocity is almost 121 .5 meters per second now as we said we can use our trigonometry to find the speed of the plane so we know that cosine of theta is equal to the adjacent over the hypotenuse and the adjacent is a hundred and twenty one point five meters per second over the hypotenuse which is the speed of the plane and we know the angle it's 53 degrees so when we solve for the velocity or the speed and we use the 53 degrees we obtain that that velocity is close to 2002 meters per second.

03:06 And that's our answer.

03:09 Now for b, we want to find a horizontal distance traveled.

03:14 And to do that, what we can do is write down that the displacement, which we're gonna call d, in the x -axis is equal to the initial velocity in the x -axis, multiply it by the time.

03:30 And we just found the speed of the airplane...

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