Daring Darless wishes to cross the Grand Canyon of the Snake River by being shot from a cannon.

Daring Darless wishes to cross the Grand Canyon of the Snake...

Key Concepts

Gravitational Acceleration

Gravitational acceleration is the constant acceleration (approximately 9.8 m/s² on Earth) that objects experience due to Earth’s gravity. It directly influences the vertical motion of a projectile and determines the time it stays in the air, making it a crucial factor in calculating projectile trajectories.

Range Equation

The range equation relates the horizontal distance traveled by a projectile to its initial velocity, launch angle, and the acceleration due to gravity. By combining the information from both the horizontal and vertical motions, the equation helps determine the necessary conditions, such as minimum launch speed, to cover a specific distance.

Initial Velocity Decomposition

Initial velocity decomposition involves breaking the launch speed into horizontal and vertical components using trigonometric functions (sine and cosine). This step is essential for separately analyzing the motion in each direction: the horizontal component, which remains constant in the absence of air resistance, and the vertical component, which is affected by gravitational acceleration.

Projectile Motion

Projectile motion is the study of the motion of an object that is thrown or launched into the air and is subject only to acceleration due to gravity. This analysis splits the motion into independent horizontal and vertical components. It is critical for understanding trajectories, such as when an object is launched at an angle and follows a parabolic path.

Transcript

00:01 We will assume that the daring, dartless is short from the cannon at the same elevation as she lands.

00:10 If we now spread the motion into two parts, t -nort to t1 and then t1 to t2 as the previous problem we have done it.

00:21 So up to the end, so obviously over there we have given x -nor over there is 0 and t -nor is obviously 0.

00:29 Y nore initial height 0 and acceleration in the wide direction is always 9 .8 meter per second square let me write it over there acceleration a is just 9 .8 meter per second square and v velocity in the velocity or v nor x will be obviously always constant and that is v1 x that is b2 x that is just v nor cause of 60 degree.

01:06 See okay and obviously we're at a square and a node x a x is obviously 0 and b nor y over there is just b node sign of 60 degree and v no v1y is 0 at the maximum height obviously it shows obvious and y2 over there is also 0 because they are coming back to the same position in the y direction as well so applying the kinetic equation for the x over there so x2 would be nothing but just x nor x nor plus v nor x into t 2 which is equals to 520 given and and yeah so 520 over there is given and so over there x nor is 0 so v nor x which is just v nor of course 60 degree and t 2 is 520 so t2 over there comes of to be 520 upon v nor cost of 60 degree that means we and now let's say this is equation one over there and for the second equation we're going to apply for the y so v1 5 would be nothing but v nor y plus a t which is a nor y or a y into t 1 which is 0 because final velocity is going to be 0 and this velocity over there is v nor of sign 60 degree and minus of 9 .08 into t1 equals to 0 by this logic and at the time in total actually over there we can say that the vertical position vertical position functional is symmetric parabola in the parameter so time t from which they deduce that t2 over there will be nothing but two times of t1 okay and this is very obvious thing you can think it takes same time like this one to open this the maximum height to goes down into the same high level that it this time and this time over there is same and that's the logic behind it so basically we can say that over there v0 sign of 60 degree minus of 9 .8 will be nothing but t 2 upon 2 and place of t 1 and we have got made a t 2 in the expression of this 1 this is 520 upon v0 equals 60 degree so v nor of sign 60 degree is equals to because when it goes to other side this will become plus so this in 9 .8 into 520 upon 2 into v.

03:38 Nor cost of 60 degree...

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