## Grantham PHY220 Week 2 Assignment Problem 8

### If a car is traveling at 50 m/s and then stops over 300 meters (while sliding), what is the coefficient of kinetic friction between the tires of the car and the road?

SOLUTION:

Draw the free-body diagram of the car

Consider the vertical direction

$\sum F_y=ma_y$

$F_N-mg=0$

$F_N=mg$

Consider the motion in the horizontal direction

Solve for the acceleration of the car.

$v^2=\left(v_0\right)^2+2a_x\Delta x$

$a_x=\frac{v^2-\left(v_0\right)^2}{2\Delta x}=\frac{0-50^2}{2\left(300\right)}=-4.17\:m/s^2$

Solve for the coefficient of kinetic friction

$\sum F_x=ma_x$

$-F_{fr}=ma_x$

$-\mu _kF_N=ma_x$

$\mu _k=\frac{ma_x}{-F_N}=\frac{m\:\left(-4.17\right)}{-m\left(9.80\right)}=\frac{4.17}{9.80}$

$\mu _k=0.43$

## Grantham PHY220 Week 2 Assignment Problem 7

### A 7.93 kg box is pulled along a horizontal surface by a force $F_p$$F_p$ of 84.0 N applied at a $47^{\circ}$$47^{\circ}$ angle. If the coefficient of kinetic friction is 0.35, what is the acceleration of the box?

SOLUTION:

The free-body diagram of the box

Solve for the normal force

$\sum F_y=ma_y$

$F_N-mg+F_psin\left(47.0^{\circ} \right)=0$

$F_N-7.93\left(9.80\right)+84.0\:sin\left(47^{\circ} \right)=0$

$F_N=16.28\:N$

Solve for the friction force

$F_{fr}=\mu _kF_N=0.35\left(16.28\:N\right)=5.70\:N$

Solve for the acceleration in the horizontal direction

$\sum F_x=ma_x$

$F_p\:cos\:\left(47^{\circ} \right)-F_{fr}=7.93\left(a_x\right)$

$84.0\:N\cdot cos\:\left(47^{\circ} \right)-5.70\:N=7.93\:kg\:\cdot \left(a_x\right)$

$a_x=\frac{51.59\:N}{7.93\:kg}=6.51\:m/s^2$

Therefore, the acceleration of the box is 6.51 m/s2 along the horizontal surface.

## Grantham PHY220 Week 2 Assignment Problem 6

### If the acceleration due to gravity on the Moon is 1/6 that what is on the Earth, what would a 100 kg man weight on the Moon? If a person tried to simulate this gravity in an elevator, how fast would it have to accelerate and in which direction?

SOLUTION:

The acceleration due to gravity on the moon is

$g_m=\frac{1}{6}\left(9.80\:m/s^2\right)=1.63\:m/s^2$

The weight of a 100-kg man on the moon is

$W_m=mg_m=\left(100\:kg\right)\left(1.63\:m/s^2\right)=163.3\:N$

If the elevator is accelerating upward then the acceleration would be greater. The person would be pushed toward the ﬂoor of the elevator making the weight increase. Therefore, the elevator must be going down to decrease the acceleration.

For a 100 kg man to experience a 163.3 N in an elevator,

$F=ma$

$163.3\:N=100\:kg\:\left(9.80\:m/s^2-a_e\right)$

$9.80-a_e=\frac{163.3}{100}$

$a_e=9.80-\frac{163.3}{100}$

$a_e=8.167\:m/s^2$

Therefore, the elevator should be accelerated at 8.167 m/s2 downward for a 100-kg man to simulate his weight just like his weight in the moon which has 1/6 of the Earth’s gravity acceleration.

## Grantham PHY220 Week 2 Assignment Problem 5

### If a 1500 kg car stopped from an in 5.6 seconds with an applied force of 5000 N, how fast was it initially traveling?

Solution:

There is a negative acceleration since the car decelerates.

$F=ma$

$a=\frac{F}{m}=\frac{5000\:N}{1500\:kg}=-3.33\:m/s^2$

$v=v_0+at$

$v_0=v-at$

$v_0=0-\left(-3.33\:m/s^2\right)\left(5.6\:s\right)$

$v_0=18.6\:m/s$

The car was initially traveling at $18.6\:m/s$.

## Grantham PHY220 Week 2 Assignment Problem 4

### A not so brilliant physics student wants to jump from a 3rd-floor apartment window to the swimming pool below. The problem is the base of the apartment is 8.00 meters from the pool’s edge. If the window is 20.0 meters high, how fast does the student have to be running horizontally to make it to the pool’s edge?

Solution:

Since the student will be running horizontally, there is no initial vertical velocity, $v_{0_y}=0$. We are also given $\Delta x=8\:m$, and $\Delta y=-20\:m$.

Consider the vertical component of the motion.

$\Delta y=v_{0_y}t-\frac{1}{2}gt^2$

$-20=0-\frac{1}{2}\left(9.80\right)t^2$

$-20=-4.9t^2$

$t^2=\frac{20}{4.9}$

$t=\sqrt{\frac{20}{4.9}}$

$t=2.02\:s$

Consider the horizontal component of the motion

$\Delta x=v_{0_x}t$

$v_{0_x}=\frac{\Delta x}{t}$

$v_{0_x}=\frac{8}{2.02}$

$v_{0_x}=3.96\:m/s$

Therefore, the student should be running 3.96 m/s horizontally to make it to the pool’s edge.

## Grantham PHY220 Week 2 Assignment Problem 1

### A ship has a top speed of 3 m/s in calm water. The current of the ocean tends to push the boat at 2 m/s on a bearing of due South. What will be the net velocity of the ship if the captain points his ship on a bearing of 55° North of West and applies full power?

Solution:

$R_x=-3\:cos\:55^{\circ }=-1.720729309\:m/s$

$R_y=3\:sin\:55^{\circ }-2=0.4574561329\:\:m/s$

The x component of the resultant is negative and the y component is positive, thus the resultant is located at the second quadrant.

$R=\sqrt{\left(R_x\right)^2+\left(R_y\right)^2}=\sqrt{\left(-1.720729309\:m/s\right)^2+\left(0.4574561329\right)^2}=1.78\:m/s$

$\theta =tan^{-1}\left(\frac{R_y}{R_x}\right)=tan^{-1}\left(\frac{0.4574461329}{-1.720129309}\right)=-14.9^{\circ}$

Therefore, the magnitude of the net velocity of the ship is 1.78 m/s, and is going 14.9 degrees North of West

## Grantham PHY220 Week 1 Assignment Problem 8

### A stone is dropped from the roof of a high building. A second stone is dropped 1.25 s later. How long does it take for the stones to be 25.0 meters apart?

Solution:

Let t be the amount of time after the first stone is dropped. The distance from traveled by the first stone is

$y_1=\frac{1}{2}gt^2$

The distance traveled by the second stone is

$y_2=\frac{1}{2}g\left(t-1.25\right)^2$

The difference between the two stones is 25.0 m after time

$y_1-y_2=25.0$

$\frac{1}{2}\left(9.80\:m/s^2\right)t^2-\frac{1}{2}\left(9.80\:m/s^2\right)\left(t-1.25\right)^2=25$

$4.9t^2-4.9\left(t^2-2.5t+1.5625\right)=25.0$

$4.9t^2-4.9t^2+12.25t-7.65625=25.0$

$12.25t=25.0+7.65625$

$12.25t=32.65625$

$t=2.67\:s$

## Grantham PHY220 Week 1 Assignment Problem 7

### Explain a possible situation where you start with a positive velocity that decreases to a negative increasing velocity while there is a constant negative acceleration.

Solution:

An example of this situation is a free-fall. If an object is thrown upward, the initial velocity is positive. Then the velocity decreases until the object thrown will reach its maximum height and then it goes back with a negative increasing velocity. In this entire flight, the acceleration is a constant negative–the acceleration due to the Earth’s gravity.

## Grantham PHY220 Week 1 Assignment Problem 6

### A sports car moving at constant speed travels 150 m in 4.00 s. If it then brakes and comes to a stop while decelerating at a rate of 6.0 m/s2 , how long does it take to stop?

Solution:

The velocity before applying the brake is

$v=\frac{150\:m}{4.00\:s}=37.5\:m/s$

Solve for t until the velocity becomes zero

$v=v_0+at$

$t=\frac{v-v_0}{a}=\frac{0-37.5\:m/s}{-6.0\:m/s^2}=6.25\:s$