# Newton's laws of motion

Problem 1:

A box of mass M is held against the ceiling by a force P acting at an angle (SYMBOL1) as shown. The block slides along the ceiling to the right at a constant speed. The coefficient of kinetic friction between the box and the ceiling is (SYMBOL2).

(a) Draw a labled free body diagram for the box

(b) Determine the force P in terms of M, g, SYMBOL1 and SYMBOL2

Problem 2:

In a James Bond movie, James in a 2000kg car driving at 10 m/s when he approaches 40m wide canyon. Rather than stopping he drives off the cliff, propelling a propeller engine which exerts a constant horizontal force of 5000N while the car is in tha air. The higher cliff is 80m above the valley and the lower cliff is 60 meters above the valley. Neglecting frictional losses, determine the speed at which he lands.

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Since the block does not fall vertically and moves with constant speed, the net force on the block is zero.

In the horizontal direction the equation is:

And in the vertical direction the equation is:

Since

We can isolate N from the vertical equation and substitute it in the horizontal equation:

Thus:

The equations of motion in the horizontal direction are:

Where the acceleration comes from the force applied by the propeller:

The vertical equations of motion are those of a free fall:

Where g is the gravitational acceleration .

To find the car's speed one has to calculate the final velocity both in the horizontal and vertical directions. For this wee need to calculate the time it took the car to jump over the chasm (if it makes it).

We can get t either from the horizontal displacement equation or from its vertical counterpart, but in any case the time found from the horizontal motion should be less than the time it took the car to fall the height difference between the cliffs (otherwise we will have a very two dimensional agent with the code name double-o-splat).

First let's calculate the horizontal acceleration:

The horizontal equation becomes a quadratic equation for t which has two roots, but we are only interested in the positive one:

From the vertical equation we get (assuming no initial vertical velocity):

Apparently Q really messed up this time, and poor ol' James will have martini on the rocks, but at least it will be shaken - not stirred. Instead of landing velocity we need to find the crash velocity.

The car will be airborne for only 2.93 seconds. After that time the horizontal velocity will be:

And the final vertical velocity is:

The speed is the magnitude of the velocity vector and is given by:

Let's hope the airbag worked......

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