PHYS1150 Problem Solving in Physics
Tutorial 1
The problems in this worksheet have a wide range of difficulties. They are specially designed to
strengthen students’ concepts and to develop the problem solving skills. Students are expected to drill
the problems with perseverance. The number of questions is more than enough to be discussed in
the tutorial class. Students can regard the remaining questions as exercises or solved problems. The
solution will be posted before the next tutorial class.
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1. If AB = 2 î − ĵ and AC = −4 î + ĵ, find BC and the angle between AB and AC .
2. Given that A(−2, −3), B(−5, −2), and C(0, 3) are the vertices of a triangle, show that 4ABC
is a right-angled triangle.
~ = −4 î − 6 ĵ − 2 k̂, B
~ = − î + 4 ĵ + 3 k̂, and C
~ = −8 î − ĵ + 3 k̂, show that they are coplanar.
3. If A
4. Show that if ~a and ~b are nonzero vectors, then |~a +~b| ≤ |~a| + |~b|. Deduce that |~a| − |~b| ≤ |~a −~b|.
Hence, show that |~a| − |~b| ≤ |~a − ~b|.
5. Given that ~a and ~b are nonzero vectors.
(a) Show that ~a − ~b and ~a + ~b are perpendicular to each other, if |~a| = |~b|.
(b) If ~a − ~b and ~a + ~b are perpendicular to each other, show that |~a| = |~b|.
6. ~a and ~b are orthogonal unit vectors and (~a − ~c) · (~b − ~c) = 0, find the maximum length of ~c.
Recall that two vectors are said to be orthogonal if they are normal to each other.
7. Given that ~a and ~b are nonzero vectors. If ~a + 3 ~b is normal to 7 ~a − 5 ~b and ~a − 4 ~b is normal
to 7 ~a − 2 ~b, find the angle between ~a and ~b.
8. A 3.0-kg object has a velocity (6.0 î − 2.0 ĵ) m/s. What is its kinetic energy at this moment?
What is the net work done on the object if its velocity changes to (8.0 î + 4.0 ĵ) m/s?
* 9. Find the time(s) between four o’clock and five o’clock at which the minute and hour hands of
a clock align to a straight line.
[Hint: Consider the position vectors of the minute and hour hands soon after four o’ clock.]
~ = î + 2 ĵ and B
~ = −2 î + 3 ĵ. Find A
~ × B,
~ the unit vector along
10. Two vectors are given by A
~
~
~
~
A × B, and the angle between A and B.
~ such that (2 î − 3 ĵ + 4 k̂) × A
~ = 4 î + 3 ĵ − 4 k̂.
11. A student claims that he has found a vector A
Do you believe this claim? Explain why or why not.
12. P is a point on the xy-plane, where OP has unit length and O is the origin as shown in the
figure. The unit vectors î and ĵ point along the x and y axis respectively. Another coordinate
system x0 y 0 is obtained by rotating the xy-coordinate axes with an angle θ counterclockwisely.
The unit vectors û and v̂ point along the x0 and y 0 axis respectively.
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(a) Express OP using the xy-coordinate system.
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(b) Express OP using the x0 y 0 -coordinate system.
(c) Express û in terms of î and ĵ. Deduce a similar expression for v̂.
(d) Show that cos(θ + φ) = cos θ cos φ − sin θ sin φ and sin(θ + φ) = sin θ cos φ + cos θ sin φ.
~ is given by q (~v × B),
~
13. The magnetic force F~ exerted on a moving charge q in a magnetic field B
~ =
where ~v is the velocity of the charged particle. A proton moves in the magnetic field B
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0.50 î T with a speed of 1.0 × 10 m/s in the directions (a) −î, (b) −ĵ, and (c) −ĵ + k̂. The
charge of a proton is 1.60 × 10−19 C, find F~ .
14. Rain is falling vertically. A man in a train travelling at 60 km/hr notices that the rain makes
lines on the window inclined at 10 degrees to the horizontal. What is the speed of the rain?
* 15. A hovercraft is 8 km due west of a ship which is sailing with a speed of 16 km/hr due north.
The speed of the hovercraft is 34 km/hr. By considering the velocity of the hovercraft
relative to the ship, find the time and the steering direction of the hovercraft such that it
can meet the ship.
[Note: It is no surprise that you can solve this problem without using the concept of relative
motion. But, the suggested approach is a brilliant idea and is full of insight!! The power of
this skill is shown in the next problem.]
* 16. This problem is modified from question 15. A hovercraft is 8 km due west of a ship which is
sailing with a speed of 16 km/hr due north. The speed of the hovercraft is 34 km/hr and the
direction is N 70◦ E. Find the shortest distance between them.
* 17. A ship is steaming due west at 18 km/hr. To an observer in the ship a hovercraft appears to
be moving in a direction north west at 12 km/hr. Find the velocity of the hovercraft.
** 18. Two particles of masses m1 and m2 and velocities ~u1 and ~u2 respectively, collide and stick
together. Assuming no external force is present, show that the energy lost in the collision can
be expressed as
1 m1 m2
|~u1 − ~u2 |2 .
2 m1 + m2
2
* 19. A force acting on a particle moving in the xy-plane is given by F~ = 2y î + x2 ĵ, where F~ is
in newtons and x and y are in meters. Four vertices of a square O(0, 0), A(0, 5), B(5, 0), and
C(5, 5) are defined on the plane and the particles moves from O to C along three paths of
straight segments:
• Path I: OAC,
• Path II: OBC,
• Path III: OC
Calculate the work done by F~ on the particle as it moves along paths I, II, and III.
R
[Hint: The work done by a force is given by F~ · d~r, where d~r = dx î + dy ĵ.]
END OF PAPER
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