Physics Vector Magnitude and Direction

Questions and Answers

What is the value of the magnitude R calculated from the vectors?

37 m/s (correct)

40 m/s

30 m/s

25 m/s

What angle θ does the resultant vector R make with the vertical?

25°

19° (correct)

12°

35°

How should the boy hold his umbrella to achieve the correct angle with the vertical?

Vertically upwards

At 19° towards the east (correct)

At 45° towards the east

At 19° facing west

What are the x- and y-components of a general vector A represented as A1 and A2?

A1 = Ax, A2 = Ay Signup and view all the answers

Which of the following describes a unit vector?

A vector of unit magnitude with no dimension Signup and view all the answers

Which method is used to resolve a vector into two component vectors?

Using unit vectors aligned along the axes Signup and view all the answers

What is true about the components Ax and Ay of vector A?

Both are real numbers and not vectors Signup and view all the answers

If vector A is resolved into A1 and A2, what can be inferred about their direction?

They lie in the same plane as A Signup and view all the answers

What does instantaneous velocity represent?

The limiting value of average velocity as the time interval approaches zero Signup and view all the answers

In the context of velocity, what does the symbol ∆r represent?

The change in position of an object Signup and view all the answers

What happens to the average velocity as the time interval ∆t approaches zero?

It approaches the instantaneous velocity Signup and view all the answers

What is the significance of the tangent line in relation to instantaneous velocity?

It shows the direction of instantaneous velocity Signup and view all the answers

In the figures representing average velocity, how are the time intervals ∆t1, ∆t2, and ∆t3 characterized?

They decrease in size Signup and view all the answers

What is the relationship between ∆t and ∆r as ∆t approaches zero?

∆r approaches zero Signup and view all the answers

If the velocity is not constant, what happens to the average velocity as you take shorter time intervals?

It can become equal to the instantaneous velocity Signup and view all the answers

Which of the following accurately describes the average velocity within a given time interval?

It is calculated by dividing displacement by time interval Signup and view all the answers

What does Equation (3.24a) represent in the context of motion in a plane?

The Law of Cosines Signup and view all the answers

How is the direction of motion expressed in relation to the resultant according to Equations (3.24e) and (3.24f)?

Through tangent and sine relationships Signup and view all the answers

What do the variables A, B, and θ represent in the context of the equations found?

Components of a vector Signup and view all the answers

In the equation for displacement, ∆r = r' – r, what does r' signify?

The final position vector Signup and view all the answers

Which of the following accurately describes the resultant of the vectors involved according to Equations (3.24e) and (3.24f)?

It involves both sine and cosine relationships Signup and view all the answers

What is the primary purpose of using vectors in describing motion in a plane based on the context provided?

To represent direction and magnitude of movement Signup and view all the answers

What does the notation r = x i + y j indicate?

A position vector in two-dimensional space Signup and view all the answers

What angle is represented by φ in the equations provided?

The angle between vector A and vector B Signup and view all the answers

What is the expression for the x-component of vector A in terms of A and the angle θ?

Ax = A cos θ Signup and view all the answers

How is the resultant vector R expressed in terms of components Rx and Ry?

R = Rx i + Ry j Signup and view all the answers

What does the formula ∆r = r' − r represent?

The displacement from one position to another Signup and view all the answers

What is the limit of average velocity v as ∆t approaches zero?

The instantaneous velocity of the object Signup and view all the answers

In the context of motion in a coordinate system, what does the vector v represent?

The tangent velocity of the object's path Signup and view all the answers

If an object's position is given by r = x i + y j, what are the components of the displacement ∆r?

∆r = (x' - x) i + (y' - y) j Signup and view all the answers

What is the relationship between Ax, Ay, and A in terms of vector components?

A = /Ax^2 + Ay^2 Signup and view all the answers

If the velocities v_x, v_y, and v_z are defined, what does their representation in unit vector notation imply?

Velocity can be represented as a combination of i, j, and k for each coordinate Signup and view all the answers

What is the net displacement of the cyclist after completing the round trip from the center O to the edge P and back to O?

0 km Signup and view all the answers

What is the average speed of the cyclist if the total distance covered during the round trip is 2 km in 10 minutes?

12 km/h Signup and view all the answers

After how many turns will the motorist's displacement be less than the total path length covered according to the described movement?

After the eighth turn Signup and view all the answers

What is the average speed of the taxi if it covers a circuitous path of 23 km in 28 minutes?

50 km/h Signup and view all the answers

How high above the ground can the cricketer throw a ball if he can throw it to a maximum horizontal distance of 100 m?

25 m Signup and view all the answers

What is the magnitude of acceleration of a stone being whirled in a circular path with a radius of 80 cm making 14 revolutions in 25 seconds?

8.0 m/s² Signup and view all the answers

What is the average velocity of the taxi over the distance of 10 km when it takes a circuitous path of 23 km?

10 km/h Signup and view all the answers

What is the maximum horizontal distance a ball can go without hitting the ceiling of a hall that is 25 m high when thrown with a speed of 40 m/s?

200 m Signup and view all the answers

Study Notes

Vector Magnitude and Direction

Resultant vector R has a magnitude of 37 m/s, calculated as ( R = \sqrt{vr^2 + vw^2} = \sqrt{35^2 + 12^2} ).
The angle θ formed with the vertical is approximately 19°, calculated using ( \tan θ = \frac{vw}{vr} = \frac{12}{35} ).
Umbrella should be held at approximately 19° with the vertical towards the east to account for wind.

Vector Resolution

A vector can be resolved into two component vectors along designated directions, labeled as λa and μb.
Unit vectors (magnitude of one) are often used for vector resolution in a rectangular coordinate system for simplicity.
General vector A can be expressed as ( A = Ax \hat{i} + Ay \hat{j} ), where:
- Ax = A cos(θ)
- Ay = A sin(θ)

Position Vector and Displacement

Position vector ( r ) of a particle in a plane is defined as ( r = x \hat{i} + y \hat{j} ).
Displacement ( Δr ) is determined by the difference between two position vectors, ( Δr = r' - r ).

Velocity and Average Velocity

Instantaneous velocity ( v ) is defined as the limit of average velocity as the time interval approaches zero: ( v = \lim_{Δt \to 0} \frac{Δr}{Δt} ).
Average velocity is represented as the direction of the tangent to the path at any point.

Vector Addition

For vectors A and B in the x-y plane, their sum R can be expressed as ( R = R_x \hat{i} + R_y \hat{j} ), where:
- ( R_x = A_x + B_x )
- ( R_y = A_y + B_y )

Displacement and Average Speed

The average speed of an object is defined as total distance divided by total time taken.
Net displacement occurs when an object returns to its starting point regardless of the path taken.

Applications

Motion analysis for a cyclist in a circular path, with calculations for net displacement, average speed, and average velocity.
Situational problems involving a cab ride compared to direct distance highlight differences in average speed versus velocity.
The maximum horizontal distance for trajectories considering height restrictions can be calculated based on speed and flight time.

Circular Motion

For projectile motions and circular motion, acceleration can be deduced from radius and rotational speed considerations.
Examples include calculating the maximum height a ball can reach based on its throw distance and the characteristics of circular motion of an aircraft.

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Description

Test your understanding of vector magnitude and direction with this quiz. It covers topics such as resultant vectors, vector resolution, and displacement in a two-dimensional plane. Perfect for those studying physics concepts in motion and forces.