Exam 3: Vectors and Motion in Two Dimensions

A child is trying to throw a ball over a fence. She gives the ball an initial speed of 8.0 m/s at an angle of 40° above the horizontal. The ball leaves her hand 1.0 m above the ground and the fence is 2.0 m high. The ball just clears the fence while still traveling upwards and experiences no significant air resistance. How far is the child from the fence?

A boulder rolls off of a very high cliff and experiences no significant air resistance. While it is falling, its trajectory is never truly vertical.

A wind farm generator uses a two-bladed propeller mounted on a pylon at a height of 20 m, as shown in the figure. The width of the pylon is very narrow, and the length of each propeller blade is 12 m. A tip of the propeller breaks off just when the propeller is vertical. The fragment flies off horizontally, falls, and strikes the ground at point P with negligible air resistance. Just before the fragment broke off, the propeller was turning uniformly, taking 1.2 s for each rotation. How far is point P from the base of the pylon?

The horizontal and vertical components of the initial velocity of a football are 16 m/s and 20 m/s respectively. If there is no air resistance, how long does it take the football to reach the top of its trajectory?

Vector $\vec { A }$ has a magnitude of 6.0 m and points 30° north of east. Vector $\overrightarrow { \mathbf { B } }$ has a magnitude of 4.0 m and points 30° east of north. The resultant vector $\vec { A }$ + $\overrightarrow { \mathbf { B } }$ is given by

A projectile leaves the ground at 150 m/s and reaches a maximum height of 0.57 km. If there was no air resistance, at what angle above the horizontal did it leave the ground?

A boat, whose speed in still water is 1.75 m/s, must aim upstream at an angle of 26.3° (with respect to a line perpendicular to the shore) in order to travel directly across the stream. (a) What is the speed of the current? (b) What is the resultant speed of the boat with respect to the shore?

The magnitude of the resultant of two vectors cannot be less than the magnitude of either of those two vectors.

An object moves in a circular path at a constant speed. Compare the direction of the object's velocity and acceleration vectors.

The sum of two vectors of fixed magnitudes has its minimum magnitude when the angle between these vectors is

An athlete competing in the long jump leaves the ground with a speed of 9.14 m/s at an angle of 35° above the horizontal. How long does the athlete stay in the air, assuming no significant air resistance?

The figure shows three vectors, $\vec { A }$ , $\overrightarrow { \mathbf { B } }$ , and $\overrightarrow { \mathrm { C } }$ , along with their magnitudes. Determine the magnitude and direction of the vector given by $\vec { A }$ - $\overrightarrow { \mathbf { B } }$ - $\overrightarrow { \mathrm { C } }$ .

For general projectile motion with no air resistance, the horizontal component of a projectile's acceleration

The figure shows four vectors, $\vec { A }$ , $\overrightarrow { \mathbf { B } }$ , $\overrightarrow { \mathrm { C } }$ , and $\vec { D }$ , having magnitudes 10.0 m, 8.00 m, 6.00 m, and 2.00 m, respectively. Find the magnitude of the sum of these four vectors.

A projectile is fired from ground level at an angle above the horizontal on an airless planet where g = 10.0 m/s². The initial x and y components of its velocity are 86.6 m/s and 50.0 m/s respectively. How long after firing does it take before the projectile hits the level ground?

Vector $\vec { A }$ has a magnitude of 7.0 m and points 30° east of north. Vector $\overrightarrow { \mathbf { B } }$ has a magnitude of 5.0 m and points 30° west of south. The resultant vector $\vec { A }$ + $\overrightarrow { \mathbf { B } }$ is given by

A runner runs on a circular path of radius 10 m. What is the magnitude of the displacement of the jogger if he runs (a) half-way around the track? (b) all the way around the track?

A 960-m wide river flows at 16 m/s as shown in the figure. Alice and John have a race in identical boats which each travel 20 m/s in still water. Alice leaves point A and steers so that she goes straight to point B directly across and then back to A. John leaves point A and steers up to point C (960 m upstream) and then returns to A. (a) Which person arrives back at point A first? (b) How much sooner than the loser does the winner arrive back at point A?

A hunter points a rifle horizontally and holds it 3.30 m above the ground. The bullet leaves the barrel at 325 m/s and experiences no significant air resistance. The acceleration due to gravity at this location is 9.80 m/s². (a) How long does it take for the bullet to strike the ground? (b) How far horizontally does it travel?

A rock is thrown from the roof of a building, with an initial velocity of 10 m/s at an angle of 30° above the horizontal. The rock is observed to strike the ground 43 m from the base of the building. What is the height of the building assuming no air resistance?