Physics 298

Sample Test 1

1.   (a)   a is a unit vector in the yz plane, having a negative z component. The y component of a is
             0.8. Evaluate the z component of a.                                                                                      (3)

     (b)   Write a in the i, j, k format                                                                                                                (2)

     (c)   What is the tangent of the angle between a and the +z axis ?                                                            (3)

     (d)   b is also a unit vector. Its y and z components are (3/7) and (-3/7), respectively.

           Calculate the x component of b.                                                                                                          (4)

     (e)   Use the dot product to evaluate the cosine of the angle between a and b.                                         (6)

     (f)   What is the magnitude of the vector given by a x b ?                                                                         (7)

Solution


2.         A large boulder is ejected vertically upward from a volcano with an initial speed of 100 m/s.  Air resistance may be ignored. [g = 10       m/s2 ]

 

a.     At what time after being ejected is the boulder moving at 30 m/s upward ?                                (5)

 

b.     At what time is it moving at 30 m/s downward ?                                                                         (5)

 

c.     When is the displacement of the boulder from its initial position zero ?                                     (5)

 

d.     When is the velocity of the boulder zero ?                                                                                   (5)

 

e.     What are the magnitude and direction of the acceleration while the boulder is moving  (i) upward ? (ii) downward ? (iii) at the highest point ?                                                                                                                                                                (2)

 

f.      Sketch ay , vy and y versus t on the diagrams below (assume that y = 0 at the moment the boulder is ejected). (3)

Solution

3).        A ball moves at a constant speed of 2 m/s on a horizontal plane following the path AOBCDEOF, where BCDE is a circular arc.  The radius of the       circular part of the path is 4 m.

 

Use the gross approximation pi = 3 throughout this problem.

 

 

 

(a)  Calculate the distance traveled by the ball along the round trip OBCDEO.                                                              (2)

 

(b) How long does it take the ball to complete this round trip ?                                                                                     (3)

 

(c)  Evaluate the magnitude of the instantaneous acceleration felt by the ball at C.  On the diagram, clearly indicate the direction of the instantaneous acceleration at C.                                                                                                                                                            (5)

 

(d) Write the instantaneous velocity at B in unit vector form.                                                                                        (2)

 

(e)  Obtain the average acceleration vector in unit vector form between points B and D.                                                (5)

 

(f)  Obtain the average velocity vector in unit vector form between B and E.                                                                 (5)

 

(g)  Calculate the average velocity over the round trip OBCDEO.                                                                                   (3)

 

Solution


4.        A soccer player takes a penalty kick towards the goal. The angle at which the ball leaves
           his/her boot is 53 with respect to the ground. The penalty spot is 30 ft from the goal and the
            cross bar is 8 ft high (g = 32 ft/s2, tan(53) = 4/3).

    (a)    If the ball hits the cross bar (i.e. it is exactly 8 ft high after traveling 30 ft), how long does it
            take to reach the bar ?                                                                                                                                                 (9)

    (b)   What must be the initial velocity of the ball, vi ?                                                                                                        (6)

    (c)    Determine the horizontal and vertical components of the velocity of the ball upon impact
            with  the cross bar.                                                                                                                                                     (6)

    (d)    Sketch these velocity components and determine the tangent of the angle the velocity vector
            makes with the vertical at the cross bar.                                                                                                                     (4)

Solution

5.         A 50 kg crate is dragged across a floor by pulling on a rope inclined 370 above the horizontal.  (g = 10 m/s2, cos(37) = 4/5)

    (a)    If the coefficient of static friction is (1/3), what minimum tension in the rope is required to
            start the crate moving ?                                                                                                                                   (12)

    (b)    If the kinetic coefficient of friction is (1/4), what is the magnitude of the initial acceleration
            of the crate ?                                                                                                                                                    (13)

Solution