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NORTHERN ILLINOIS UNIVERSITY

PHYSICS DEPARTMENT

Physics 253 – Basic Mechanics Fall 2016

Problem Set #1

Problem Set Due: Wed., Aug. 29, 2016

Read Lecture Notes #1 & #2 (Giancoli: Chapters 1&2)

Remember to write all answers with the proper # of significant figures

Show explicitly all your work for full credit

Section 2.2:

1. Express the position vector, �⃗� , in terms of (i) unit vectors, and (ii) magnitude and angle for the two cases below:

(a) Point 𝑃 is located at the coordinate (−3.0,4.0).

(b) Point 𝑃 is located at the coordinate (-1.25,-4.6).

Make certain you express your solutions with the proper # of significant figures.

𝑥

𝑦

𝑟

𝑃

𝑥

𝑦

𝑃

𝑟

Section 2.3:

2. An object’s position changes with time according to the graph shown below. For the positions located at Points A, B, C, state whether the velocity is positive, negative, or

zero by drawing the tangent lines to the curve and examining their slopes.

(a) At which point is the object moving the slowest?

(b) At which point is the object moving the fastest?

(c) At points A, B, C, state whether the velocity is positive, negative, or zero by drawing the tangent lines to the curve and examining their slopes.

𝑥

𝑡

𝐴 (𝑎𝑡 𝑡ℎ𝑒 𝑣𝑒𝑟𝑦 𝑡𝑜𝑝 𝑜𝑓 𝑝𝑒𝑎𝑘)

𝐵

𝐶

3. The derivative of a polynomial function, n t , is defined as:

1 n

ndt nt

 

Example: The position of a particle at time t is given by   37.8 9.2 2.1x t t t  

Its velocity will then be 0 1 1 1 3 1

0 7.8 1 9.2 3 2.1 dx

v t t t dt

         

0 9.2 6.3t  

(a) Find the derivative of 2 3 4

0 1 2 3 4    x a at a t a t a t where 0a , 1a , 2a , 3a ,

and 4a are constants.

(b) Find the derivative of 1 2 3 4

0 2 3 4     

a a a a x a

t t t t where 0a , 1a , 2a , 3a , and 4a

are constants.

Section 2.4:

4. The position of a particle at time t is given by   210 0 2x t x v t at   where 0x , 0v , and a are constants independent of time.

(a) What is the velocity of the particle at time t ?

(b) What is the acceleration of the particle at time t ?

5. The position of a particle moving on the x-axis is given by

. 𝑥 = (6.5𝑚) + (3.7𝑚/𝑠𝑒𝑐)𝑡 − (0.2𝑚/𝑠𝑒𝑐2)𝑡2

with x in meters and t in seconds.

(a) What is the velocity of the particle at 𝑡 = 2.6 sec?

(b) Is the velocity constant or does it change with time?

Section 2.5:

The integral of a polynomial function, n t , is defined as:

1 11

1 1 1

f f

i i

t n nnt fn i

t t

t tt t dt

n n n

     

    .

Example: The velocity, dx

v dt

 , of a particle at time t is given by

  3

7 8 9 2 2 1. . .v t t t   Its position can then be found by integrating both sides of the equation:

dx vdt

f f

i i

x t

x t dx vdt 

 0 0 1 37 8 9 2 2 1. . .f f i i

x t

x t x dx t t t dt   

0 1 0 1 1 1 3 1

7 8 9 2 2 1 0 1 0 1 1 1 3 1

. . . f f

i i

x t

x t t t          

       

     2 2 4 4 9 2 2 1

7 8 2 4

. . .

i i i if f f f x x t t t t t t      

which we write as:

𝑥 − 𝑥0 = 7.8𝑡 + 9.2

2 𝑡2 −

2.1

4 𝑡4

since we always set 𝑥𝑓 = 𝑥, 𝑥𝑖 = 𝑥0, 𝑡𝑓 = 𝑡, 𝑡𝑖 = 0

6. The acceleration, 𝑎𝑥, of a particle is constant (independent of time). (a) By integration, show that the velocity of the particle is 𝑣𝑥 = 𝑣𝑥0 + 𝑎𝑥𝑡 where

𝑣𝑥0 is the initial velocity (let 𝑡𝑓 = 𝑡 and 𝑡𝑖 = 0, and note that 𝑎𝑥 = 𝑑𝑣

𝑑𝑡 ).

(b) By integration, find the position of the particle at time t . (let 𝑥𝑓 = 𝑥, 𝑥𝑖 = 𝑥0, 𝑡𝑓 = 𝑡, and 𝑡𝑖 = 0)? (Note: the solutions to Part (a) and (b) are given in the Lecture notes)

7. A particle’s acceleration along the 𝑥-axis is 𝑎𝑥 = 2.3 + 5.2𝑡, with 𝑡 in seconds and 𝑎𝑥 in meters per second squared.

(a) By integrating, what is the velocity of the particle at time 𝑡?

(b) What is the position of the particle at time 𝑡?