Question

1. Given two vectors such as

A⃗ =2x̂ +4ŷ +7ẑ

B⃗ =−3x̂ +8ŷ −2ẑ

what is their sum? What is its length, and what angle does it make to the plane of x and y? What is the dot product A⃗ ⋅B⃗

? What is the dot product of their sum with a unit vector in the z direction (A⃗ +B⃗ )⋅ẑ

? (Hint: this gives the z-component of the sum.).

2. When you apply a force of 100 N to a 25 kg box resting on a table, what is the smallest coefficient of static friction that will give a frictional force sufficient to prevent the box from moving? Suppose you push just a little harder and it starts to move. If the coefficient of kinetic friction is 0.1, how fast will it be going after 2 seconds of pushing?

3. If you walk up a mountain that is 1.5 km high (1500 meters), and your mass is 65 kg (your weight is about 143 lb of gravitational force at Earth's surface), how much energy do you use in joules to overcome Earth's gravity? How much would that be in food calories, where 1 calorie is 4.18 J and a "food" calorie is a kilocalorie in the units used by physicists and chemists. Similarly, if an object of your falls from low Earth orbit to Earth's surface, a distance of say 400 km which is the approximate height of the International Space Station orbit, how much kinetic energy would it have on reaching the Earth's surface -- assuming (a) it starts from rest, and (b) it does not lose energy to the atmosphere on the way down.

Answer 1

1. The sum is A+B=2x-3x+4y+8y+7z-2z

=12y-x+5z

Length= √(12²+1²+5²)= 13.038 units.

Let angle be A. Since X-Y plane is represented z with coordinates (0,0,1). Let vector A+B be C.

Then, z. C= zCsin A

Since only z coordinates remain, L.H.S = 5 and R.H.S product of magnitudes, is 1× 13.038 sinA

Hence, 5= 13.038 sin A

so, A= 22.55 degrees

The dot product A.B = (2x+4y+7z).(-3x+8y-2z)

= -6+32-14= 12 units

Unit vector Z= z

Hence, (A+B).Z =( 12y-x+5z).(z)= 0-0+ 5= 5 units.

2. Static friction is the minimum force that needs to be overcome to set the body to motion from rest. Kinetic friction is the force that acts between bodies when the body is moving, in an opposite direction to motion. Please refer to the image attached.

3. Mass=65kg and height h=1500m

Energy used to overcome gravity is potential energy mgh