Problem 3-6. A dart is thrown upward at an angle of 25^o above the vertical. The vertical component of the dart's velocity is v_y = 2.2 m/s. Determine the x component of the velocity.

The components of the velocity are related to the angle by

Problem 3-15. The punter on a football team tries to kick a football so that it stays in the air for a long "hang time". If the ball is kicked with an initial velocity of 25.0 m/s at an angle of 60.0^o above the ground, what is the "hang time"?

The initial vertical component of the velocity is

The hang-time is found by setting y = 0 in one of the equations for the vertical displacement

Problem 3-28. A horizontal rifle is fired at a bull's-eye. The muzzle speed of the bullet is 670 m/s. The barrel is pointed directly at the center of the bull's eye, but the bullet strikes the target 0.025 m below the center. What is the horizontal distance between the end of the rifle and the bull's-eye?

During its flight, the bullet falls a vertical distance of .025m. This takes a time

In this time, the bullet has traveled a horizontal distance

Problem 3-47. A small can is hanging from the ceiling. A rifle is aimed directly at the can. At the instant the gun is fired, the can is released. Ignore air resistance and show that the bullet will always strike the can, regardless of the initial speed of the bullet. Assume that the bullet strikes the can before the can hits the ground.

Suppose that the bullet is initially a horizontal distance R and a vertical distance H from the can. Then, the since the bullet is aimed directly at the can, the vertical and horizontal components of its initial velocity are related by

If the bullet is to hit the can, it must still travel the same horizontal distance, so the time of flight is

using the previous result. At the end of this time, the vertical displacement of the bullet is

But the can has also fallen a distance ˝gt² in this time, and so the bullet hits the can.

Problem 3-54. A remote-controlled model airplane is flying due east in still air. The airplane travels with a speed of 22.6 m/s relative to the air. A wind suddenly begins to blow from the north toward the south with a speed of 8.70 m/s. Find the velocity (magnitude and direction) of the airplane as seen by the controller standing on the ground. Determine the directional angle relative to due east.

Velocity of plane relative to ground = velocity of plane relative to air + velocity of air relative to ground

= 22.6 m/s due east + 8.70 m/s towards south.

The magnitude of the velocity is

and the velocity is directed south of east by an angle of