You encounter the two vectors below. Find the scalar product M · N for the two vectors M and N.

Given î, ĵ and k̂ represent unit vectors along the +x, +y, and +z axes respectively. Determine the cross products of the following: ĵ × î, k̂ × î, and k̂ × ĵ.

Given two vectors, M = 5.00 i + 4.00 j and N = 3.00 i − 7.00 j, find the vector product M × N (in unit vectors). What is the magnitude of the vector product? 

At t= 0, a submarine is moving at v = -8.0 ĵ m/s and is located at r₀ = 5000 î + 125 ĵ + 750 k̂  ( in meters) with respect to a reference island. At t= 0 s, the submarine accelerates at a rate of a = 1.25  î - 2.50  ĵ (m/s²) for 2 minutes. Calculate the submarine's position vector after 2 minutes of motion.

A small disk slides on a very smooth horizontal surface. The variations of the horizontal and vertical components of the disk's velocity are represented in the figure below. At time t = 0, the disk is at the origin. Calculate the disk's position after 2.0 s.

A stone is thrown horizontally from the top of a building with an initial speed of 12 m/s. It takes 4.0 seconds for the stone to hit the ground below. Calculate i) the height of the building, and ii) the horizontal distance from the base of the building to where the stone hits the ground.

A steel block, with a mass of 4.5 kg, and a concrete block, with a mass of 5.5 kg, are positioned on a frictionless inclined plane as shown. The concrete block is initially 0.90 m above the floor. The falling concrete block causes the steel block to accelerate at 5.4 m/s² on the frictionless surface. Given that the system starts from rest and air resistance is negligible, calculate the velocity of the concrete block just before it makes contact with the floor.

A helicopter is traveling westward at an airspeed of 125 km/h. A crosswind from the northeast begins to affect the helicopter, with an average velocity of 35 km/h. In what direction should the helicopter's nose be pointed to minimize lateral drift and maintain a westward course?

A gardener is watering his plants on sloppy land inclined at 20° as shown in the figure below. The water is leaving the pipe at an angle of θ° with respect to the slope. Determine the angle at which the water should leave the pipe so that it covers the maximum distance 'd'.

Hint: use the following expression d/dθ(A•cosθ•sinθ - B•sin²θ) = A•cos(2θ) - B•sin(2θ)

Rocks on a slippery mountain can turn into swift projectiles as they slide. Imagine a rock at the ridge of a mountain incline at a 22° angle below the horizontal, with an initial velocity of 6.0 m/s. If the mountain's edge is 13 m above the ground, calculate the horizontal distance between the base of the mountain to the point at which the rock hits the ground.

A thief steals some precious diamonds preserved as ancient treasures in a museum. The security comes to know about the robbery and they carry out a search operation for the stolen diamonds. The thief finds himself locked inside as all the entrances and exits are sealed. In order to save himself from getting caught along with the diamonds he has to throw the diamond pouch to his partner who is outside the gate as shown in the figure. Determine the angle at which the pouch needs to be thrown so that it safely reaches the partner thief.

A 5-kg tool bag is attached to a pulley system in order to transport it from the ground floor to the higher floors of the construction site with an upward velocity of 12 m/s. It contains a 3-kg toolbox. For some unknown reason, it opens and a 500-g spring launches itself from inside it with an initial velocity of 10 m/s perpendicular to the path of the ascending tool bag. 3 seconds later, this spring hits the ground. Assume that the tool bag continues its rise at a constant speed of 12 m/s. Upon hitting the floor, how far is the spring from the tool box?

During a football game, a player kicks the ball, initially at rest, creating a high arc across the field. Observations reveal that the total time the ball remains airborne from kick to landing is 4.5 seconds. Calculate the maximum height $h_{\max }$ the ball reaches, using its total time of flight.

A water balloon is released from a stationary hot air balloon that is 100 meters above the ground as shown below. The water balloon is released with an initial velocity of 15.0 m/s at an angle of 30.0° from the horizontal. Calculate the magnitude of the velocity of the water balloon at the instant it lands.

Ramps are useful machines when loading trucks. In one instance, a worker pulls on a trolley being loaded on a truck with an upward force F that has a direction of 28.0° with the ramp. If the ramp has a slope angle of 15.0°, what is the value of the component F_y perpendicular to the ramp when the F_x component parallel to the ramp is equal to 86.0 N?

Draw the following diagrams i) motion diagram ii) force identification diagram iii) free-body diagram for a stone thrown (already in the air) vertically upward that experiences significant air resistance.

As you near an intersection on your bicycle, which weighs 12 kg, traveling at a speed of 8.0 m/s, you find yourself 30 m from the intersection just as the traffic light switches to yellow. Anticipating that the yellow light will last for 5.0 s, and considering the intersection spans 20 m across, you decide to increase your pedaling effort to ensure a safe passage. Considering your mass of 60 kg and your ability to apply a forward force of 150 N through pedaling, can you successfully navigate the entirety of the intersection before the light turns red?

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Two patients X (60kg) and Y(85kg) are in an elevator descending at a rate of 15m/s. Patient X exits the elevator on the sixth floor. Suppose it takes the elevator 8s to stop on the 6th floor, calculate the weight of both patients before the elevator begins to slow down.

In a warehouse, a 30.0-kg wooden crate is placed on a sturdy table. A 20.0-kg metal box is then placed on top of the wooden crate as shown. Determine the normal force that the table exerts on the wooden crate and the normal force that the wooden crate exerts on the metal box.

Bob observes his phone dangling from a cord while the train he is in accelerates out of the station, lasting about 18 s. If the cord forms an angle of 26° with respect to the vertical during this acceleration, estimate the speed of the train at the end of the 18 s.

At a carnival, a challenge is set up for two people. To win the challenge they must pull a rectangular box to the finish line but both of them should have equal pulls. The angle between the two ropes tied to the box is 30 degrees. If the friction force on the box is 970 N, determine how much force each person must use to pull the box at a constant 3.0 m/s.

During a rescue mission, three firefighters are linked together by a rope as they ascend a steep slope. The slope is inclined at 35° to the horizontal, as shown in the diagram. Unexpectedly, the last firefighter loses footing, leading the second firefighter to also lose balance. Fortunately, the first firefighter successfully supports both of them. With each firefighter weighing 70 kg, determine the tension in each of the two segments of the rope connecting the three firefighters. Assume negligible friction between the slipping firefighters and the surface.

An industrial worker needs to move a 75-kg crate across a concrete floor. The coefficient of kinetic friction between the crate and the concrete floor is μₖ = 0.31. The worker applies a pulling force of 255N at an angle of 6.0° from the horizontal to slide the crate. Considering the setup, explain why the crate experiences a different acceleration when the force is applied at an angle compared to when the force is applied horizontally.

A 31.0 N crate of feeds is resting on a horizontal floor. The coefficients of static and kinetic friction between the crate and the floor are 0.52 and 0.34 respectively. A he-goat is pushing against the crate applying a horizontal force on the crate. Determine the least horizontal force applied by the he-goat to move the crate.

A small metallic box is projected up along a smooth inclined plane with a speed of 2.0 m/s. The length of the incline is 20 cm, and its height is 5.0 cm. Calculate the box's speed at the summit of the incline.

The figure shows an iron slab of mass 4.0 kg attached to a box having a mass of 5.0 kg via a massless string. The pulley connecting them has a mass of 150 g and a radius of 10 cm. Calculate the magnitude of acceleration experienced by the iron slab if it has a force of 25 N applied toward the left. Also, calculate the magnitude of tension(s) in different parts of the string. Consider the pulley and the horizontal surface to be frictionless.

The figure shows a 250 g slab placed on a 950 g frictionless wedge structure. The whole assembly rests on level ground. Determine the normal reaction experienced by the wedge if the slab was sliding downwards on the inclined structure.