A cannon with a muzzle speed of 1000 m / s is used to start...

Key Concepts

Projectile Motion

Projectile motion is the study of objects that are launched into the air and move under the influence of gravity. It involves analyzing the path of the object (its trajectory) by considering the horizontal and vertical components of its motion separately, using kinematic principles.

Decomposition of Initial Velocity

Decomposition of initial velocity involves breaking a projectile's speed into horizontal and vertical components using trigonometric functions. This separation is essential because the horizontal motion is constant while the vertical motion is accelerated by gravity, allowing us to analyze each independently.

Kinematic Equations for Uniform Acceleration

Kinematic equations describe the relationships between displacement, velocity, acceleration, and time in motion scenarios. In projectile motion, these equations are applied separately to horizontal (constant velocity) and vertical (constant acceleration) movements, forming the basis for predicting the projectile’s path.

Trajectory Equation

The trajectory equation relates the vertical displacement of a projectile to its horizontal displacement, initial velocity components, and gravitational acceleration. It is used to determine where a projectile will be at a given time or to find the conditions (such as launch angle) required to reach a specific target.

Algebraic Problem-Solving in Projectile Motion

Solving projectile motion problems often requires setting up equations from the separate horizontal and vertical motions, eliminating the time variable, and sometimes solving a quadratic equation in terms of the launch angle or another variable. This approach combines algebraic manipulation and trigonometric identities to resolve the problem.

Transcript

00:01 Hi, in this question we have a cannon that can fire bullets with maximum speed of 1 ,000 meters per second.

00:12 And this happened to be on a mountain, so the sniper is fixed here, and the target is up on the mountain at 800 meters high.

00:25 And it's 2 ,000 meter far horizontally speaking.

00:32 We're asked to calculate the angle at which the bullet should be fired in order to hit this target.

00:43 So we have first x is equal to 2 ,000 meter and we have y, which is the height 800 meters.

00:55 And this is simply a point on that trajectory we're looking for.

01:00 So we assume that there should be two solutions for this end.

01:06 That's because it can happen for some point to exist on different trajectories at the same time.

01:17 So let's start first by applying the equation for x.

01:21 We know that x is equal to v, cosine, theta, multiplied by t.

01:30 Applying this here, we know that also the maximum muzzle speed is 1 ,000 meters per second.

01:39 So that's 1 ,000 multiplied by cosine theta that we're looking for, multiplied by t, and that's equal to 2000.

01:56 From this equation, we can get that t is equal to 2 .2.

02:03 Divided by cosine theta.

02:07 And let's put a box here because we will use this to substitute back in the second equation for y.

02:17 Now let's substitute in the equation of y.

02:21 Y is equal to v sine theta t minus half multiplied by 9 .8 multiplied by t squared.

02:35 This is the equation of kinematics for projectile motions.

02:40 Let's substitute back.

02:42 The height is 800 and the velocity is 1 ,000 and we have sign theta that we want to know and t minus half multiplied by 9 .8, that's 4 .9 t squared.

03:01 Let's now substitute for t using this equation in the red box here.

03:08 So now we will have 800 equal to 2 multiplied by 1000, that's 2 ,000, sine theta divided by cosine theta, that's the tangent of theta...

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