What is the time of flight for the projectile?
Time of flight = (2 * initial speed * sin(angle)) / g.
p.3
Applications of Motion Concepts
What will hard work in engineering physics lead to?
Grasping knowledge of engineering physics and developing critical thinking skills.
p.35
Uniform Circular Motion
What is true about the acceleration in uniform circular motion?
It is constant in magnitude but continuously changing in direction.
What is the restriction on the relative velocity between observers S and S'?
It must be constant in both magnitude and direction.
How is the position of particle P indicated according to system S?
By the vector ๐ ๐ท๐บ.
p.34
Uniform Circular Motion
What is the formula for average acceleration in uniform circular motion?
a = โv / โt = 2v sin(ฮธ/2) / r.
p.8
Acceleration in Circular Motion
What are the components of instantaneous acceleration?
๐๐ = ๐
๐๐ / ๐
๐, ๐๐ = ๐
๐๐ / ๐
๐, ๐๐ = ๐
๐๐ / ๐
๐
What is the goal of the contestant in the projectile motion problem?
To drop a package on a target.
What formula is used to find the time of fall in projectile motion?
y = (vโ sin โ
โ)t - (1/2)gtยฒ.
What is the maximum range for a given initial speed?
The maximum range occurs at an angle of 45 degrees.
What angle does the soccer player kick the ball?
36 degrees from the horizontal.
p.5
Position, Velocity, and Acceleration
How is average velocity defined?
Average velocity is given by ๐ = โ๐ / โ๐.
What does the notation ๐
๐ ๐ท๐บ ๐
๐ represent?
It represents the rate of change of position of P as measured by S over time.
p.13
Motion with Constant Acceleration
What is the scalar equation for the x-component of velocity?
๐โ = ๐โโ + ๐โ๐ก
p.13
Motion with Constant Acceleration
What is the equation for position with constant acceleration?
๐ = ๐โ + ๐โ๐ก + rac{1}{2}๐๐กยฒ
p.35
Uniform Circular Motion
Where is the velocity vector directed in uniform circular motion?
Always tangent to the circle in the direction of motion.
What angle of sight ฮฑ should the package be released to strike the target?
The specific angle is not provided in the text; it requires calculation based on the given parameters.
What is the formula for the range R in projectile motion?
R = (vโยฒ / g) * sin(2ฮธโ)
In which dimensions is the motion considered for simplicity?
In two dimensions, specifically the xy plane and x'y' plane.
What is the equation for relative motion?
๐ ๐ท๐บ = ๐ ๐ท๐บโฒ + ๐ ๐บ โฒ ๐.
What does the vector ๐ ๐ โฒ ๐ represent?
The position of the S' system with respect to the S system.
Under what condition do all observers conclude that a particle moves with constant velocity?
If they are in frames of reference that move at constant velocity with respect to the first observer.
What is the significance of the equation ๐ ๐ท๐บ = ๐ ๐ท๐บโฒ?
It indicates that the acceleration of P measured by the two observers is identical.
p.8
Acceleration in Circular Motion
Does the separation of components of acceleration affect each other?
No, for example, ๐๐ has no effect on ๐๐ and ๐๐.
How do you find the time at which the ball reaches the highest point of its trajectory?
Use the formula t1 = (initial speed * sin(angle)) / g, where g is the acceleration due to gravity.
p.8
Acceleration in Circular Motion
What does โ๐ represent?
A vector having both direction and magnitude.
What is the formula to find the maximum height of the projectile?
Maximum height = (initial speed^2 * (sin(angle))^2) / (2 * g).
p.34
Uniform Circular Motion
What does the average change in velocity depend on in uniform circular motion?
It depends on the angle ฮธ and the radius r.
p.39
Uniform Circular Motion
Why is the centripetal acceleration equal to g in this context?
Because the satellite is in free fall towards the Earth while moving in a circular path.
p.18
Motion with Constant Acceleration
What is the value of constant acceleration in projectile motion?
g (acceleration due to gravity), directed downward.
p.31
Uniform Circular Motion
What defines uniform circular motion?
A particle moves at constant speed in a circular path.
How can the time tโ at which the ball returns to the ground be found?
Using the equation y = (vโ sin ฮธโ)t - (1/2)gtยฒ.
What is the significance of Eq. C in motion concepts?
Eq. C is often called the Galilean form of the law of transformation of velocities.
p.33
Uniform Circular Motion
What is the formula for the arc length traversed during a time interval โt in uniform circular motion?
Arc length P1P2 = rฮธ = vโt.
What form does the trajectory equation of a projectile take?
๐ฆ = ๐๐ฅ โ ๐๐ฅยฒ, indicating a parabolic trajectory.
p.8
Acceleration in Circular Motion
Can a change in the direction of velocity produce acceleration?
Yes, even if the magnitude of the velocity does not change.
What remains constant throughout the flight in projectile motion?
The horizontal component of the velocity vector.
p.32
Uniform Circular Motion
How do the magnitudes of v1 and v2 compare?
They have the same magnitude v (constant speed).
What is the formula to calculate the maximum height (y_max) in projectile motion?
y_max = (v_{0} sin โ
_{0}) t - (1/2) g tยฒ.
What is the formula for calculating the horizontal distance traveled by a projectile?
๐ = (๐โ cos โ
โ) ๐
What is the formula for the range R of a projectile?
R = (2 * vโยฒ * sin(ฮธโ) * cos(ฮธโ)) / g = (vโยฒ * sin(2ฮธโ)) / g.
p.7
Position, Velocity, and Acceleration
How can instantaneous velocity be expressed in terms of its components?
๐ = ๐๐๐ข + ๐๐๐ฃ + ๐๐๐ค.
How is the angle of sight (ฮฑ) calculated in projectile motion?
ฮฑ = tanโปยน(๐/๐).
What does the equation ๐
๐ ๐ท๐บ / ๐
๐ = ๐
๐ ๐ท๐บโฒ / ๐
๐ + ๐
๐ ๐บ โฒ ๐ / ๐
๐ represent?
The relationship between the velocities of different observers over time.
Does the time of fall depend on the speed of the plane?
No, it does not depend on the speed of the plane.
What does the derivative form of Eq. B represent?
It represents the relationship between the rates of change of positions of P, S, and Sโ over time.
How is the magnitude of the resultant velocity vector at any instant calculated?
๐ = โ(๐ ๐ยฒ + ๐ ๐ยฒ).
How do you calculate the range of the projectile?
Range = (initial speed^2 * sin(2 * angle)) / g.
p.32
Uniform Circular Motion
What is the difference between v1 and v2 in uniform circular motion?
Their directions are different.
How is the direction of the final velocity determined?
Using tan(ฮธ) = v_y / v_x
p.35
Uniform Circular Motion
What is the name given to the inward-directed acceleration in uniform circular motion?
Radial or centripetal acceleration.
What is the relationship between observers S and S' in relative motion?
They have their own reference frames with attached Cartesian coordinate systems.
Is there a horizontal component of acceleration in projectile motion?
No, there is no horizontal component of acceleration.
Does the restriction on relative velocity include the motion of particle P?
No, it does not include the motion of particle P being observed by S and S'.
What is the relationship expressed in Eq. C regarding the velocity of P?
The velocity of P as measured by S is equal to the velocity of P as measured by Sโ plus the relative velocity of Sโ with respect to S.
How is the position of particle P indicated according to system S'?
By the vector ๐ ๐ท๐บโฒ.
What is the relationship between the velocities of the plane with respect to the ground and the air?
๐ ๐ท๐ฎ = ๐ ๐ท๐จ + ๐ ๐จ๐ฎ.
What is the speed of the wind affecting the airplane?
65 km/h (blowing due north).
p.32
Uniform Circular Motion
What are P1 and P2 in the context of uniform circular motion?
They are the positions of the particle at times t1 and t2 respectively.
p.33
Uniform Circular Motion
Where does the change in velocity โv point in uniform circular motion?
Toward the center of the circle.
p.5
Position, Velocity, and Acceleration
What does the time interval โ๐ represent?
The difference between two time points, โ๐ = ๐โ - ๐โ.
p.13
Motion with Constant Acceleration
What is the equation for average velocity in terms of initial and final velocities?
๐ = ๐โ + rac{1}{2}(๐โ + ๐)๐ก
Why is the final speed equal to the initial speed in projectile motion?
Because energy is conserved in the absence of air resistance.
p.6
Position, Velocity, and Acceleration
What happens to the vector โ๐ as the interval โ๐ is reduced?
It approaches the actual path and becomes tangent to the path in the limit โ๐ โ 0.
p.12
Motion with Constant Acceleration
What happens to the components of acceleration in constant acceleration motion?
The components of acceleration do not vary.
p.31
Uniform Circular Motion
How do velocity and acceleration behave in uniform circular motion?
Both are constant in magnitude but change their directions continuously.
What are the values of tโ and tโ in the given problem?
tโ = 0.93 s and tโ = 1.86 s.
What does the vertical velocity component represent in projectile motion?
It represents that of free fall.
How is the magnitude of the ground velocity (ground speed) calculated?
๐ ๐ท๐ฎ = โ(๐ ๐ท๐จยฒ + ๐ ๐จ๐ฎยฒ).
p.9
Position, Velocity, and Acceleration
What is the position of the particle at t = 3 s?
x(3) = 3^3 - 32(3) = -75 m and y(3) = 5(3^2) + 12 = 63 m.
If the pilot wishes to fly due east, what must be the heading?
The compass must read approximately 28.5 degrees south of east.
p.32
Uniform Circular Motion
What does the velocity vector at P2 represent?
It is a vector tangent to the curve at P2, denoted as v2.
How does the horizontal velocity component behave throughout the flight?
It retains its initial value.
What is the direction of the final velocity in this scenario?
36ยฐ clockwise from the x-axis.
p.36
Uniform Circular Motion
Is there a fixed relation between the directions of acceleration and velocity in uniform circular motion?
No, there is no fixed relation in general.
p.6
Position, Velocity, and Acceleration
What does the average velocity approach as โ๐ approaches 0?
The instantaneous velocity.
What is the definition of the horizontal range R of a projectile?
The distance along the horizontal where the projectile returns to the level from which it was launched.
p.11
Acceleration in Circular Motion
What does the acceleration vector's direction relate to?
It has no particular relationship to the direction of either position or velocity vectors.
p.5
Position, Velocity, and Acceleration
How is the position of a particle in three dimensions represented?
By the vector ๐ = ๐๐ข + ๐๐ฃ + ๐๐ค in Cartesian coordinates.
p.8
Acceleration in Circular Motion
How is instantaneous acceleration defined?
๐ = lim (โ๐ โ 0) โ๐ / โ๐ = ๐
๐ / ๐
๐
What observation can be drawn from tโ and tโ?
tโ is double tโ, indicating the time of ascent and descent are equal in projectile motion.
p.5
Position, Velocity, and Acceleration
What does the vector โ๐ represent?
The displacement (change in position) of the particle during the time interval โ๐.
What does the variable g represent in the range formula?
The acceleration due to gravity.
p.5
Position, Velocity, and Acceleration
What is the relationship between average velocity and displacement?
Average velocity ๐ has the same direction as the displacement vector โ๐.
What are the components of the initial velocity vector at t = 0?
๐ ๐๐ = ๐ ๐ ๐๐๐ โ
๐ and ๐ ๐๐ = ๐ ๐ ๐๐๐ โ
๐.
p.33
Uniform Circular Motion
What is the relationship between arc length, radius, and angular displacement in uniform circular motion?
Arc length = radius ร angular displacement (s = rฮธ).
p.9
Position, Velocity, and Acceleration
How do you calculate the velocity of the particle?
Velocity is the derivative of position with respect to time: v(t) = dx/dt and v(t) = dy/dt.
p.9
Position, Velocity, and Acceleration
What is the velocity of the particle at t = 3 s?
v_x(3) = 3^2 - 32 = -23 m/s and v_y(3) = 10(3) = 30 m/s.
How do you find the velocity when the ball strikes the ground?
Use the formula for final velocity: v = sqrt(initial speed^2 + 2 * g * maximum height).
p.35
Uniform Circular Motion
What remains constant in uniform circular motion?
The magnitude of velocity (v).
p.4
Applications of Motion Concepts
What are the components of the assessment?
Assignment 1, Assignment 2, Test 1, Test 2.
p.6
Position, Velocity, and Acceleration
What is true about the vector ๐ in relation to the path of the particle?
The vector ๐ is tangent to the path of the particle at every point in the motion.
p.7
Position, Velocity, and Acceleration
What is the formula for instantaneous velocity?
๐ = ๐
๐/๐
๐.
At t = 6.78s, what is the horizontal distance traveled by the package?
๐ = 0.292 km or 292 m.
p.31
Uniform Circular Motion
Can you give examples of uniform circular motion?
Earth satellites, points on spinning rotors, phonograph records, and computer discs.
What happens when you set y = 0 and x = 0 in the projectile motion equation?
It allows you to derive the range of the projectile.
What does Eq. B state about the position of P?
The position of P as measured by S is equal to the position of P as measured by Sโ plus the position of Sโ as measured by S.
p.31
Uniform Circular Motion
How do humans participate in uniform circular motion?
We participate due to the rotation of the Earth.
p.7
Position, Velocity, and Acceleration
Under what condition can two vectors be considered equal?
Their corresponding components must be equal.
What is the equation for the vertical velocity component in projectile motion?
๐ ๐ = ๐ ๐๐ + ๐ ๐ ๐ = ๐ ๐ ๐๐๐ โ
๐ โ ๐๐.
What is the equation for the x-coordinate of a particle's position in projectile motion?
๐ฅ = ๐ฅโ + ๐ฃโโ๐ก + ยฝ๐โ๐กยฒ = (๐ฃโ cos โ
โ)๐ก
What is the equation for the y-coordinate of a particle's position in projectile motion?
๐ฆ = ๐ฆโ + ๐ฃแตงโ๐ก + ยฝ๐แตง๐กยฒ = (๐ฃโ sin โ
โ)๐ก โ ยฝ๐๐กยฒ
p.34
Uniform Circular Motion
How is instantaneous acceleration defined in uniform circular motion?
a = lim (โt โ 0) โv / โt = vยฒ / r lim (โt โ 0) sin(ฮธ/2) / (ฮธ/2).
How do you calculate the velocity of the plane with respect to the ground?
By using vector addition of the airplane's velocity and the wind's velocity.
What are the values used to calculate the ground speed of the plane?
๐ ๐ท๐จ = 215 km/h and ๐ ๐จ๐ฎ = 65 km/h.
p.36
Uniform Circular Motion
What is a key characteristic of uniform circular motion regarding acceleration?
There need not be any motion in the direction of an acceleration.
p.12
Motion with Constant Acceleration
What characterizes motion with constant acceleration?
The acceleration vector does not vary in magnitude or direction.
p.6
Position, Velocity, and Acceleration
What is the formula for instantaneous velocity?
๐ = lim โ๐ โ 0 (โ๐ / โ๐) = ๐
๐ / ๐
๐.
p.11
Acceleration in Circular Motion
What is the total acceleration at t = 3?
a = 18i + 10j in units of m/sยฒ.
p.7
Position, Velocity, and Acceleration
What are the scalar equations corresponding to the components of instantaneous velocity?
๐๐ = ๐
๐/๐
๐, ๐๐ = ๐
๐/๐
๐, ๐๐ = ๐
๐/๐
๐.
What values are used to calculate the angle of sight in this scenario?
๐ = 292 m and ๐ = 225 m.
What is the formula used to calculate the pilot's ground speed?
๐ ๐ท๐ฎ = ๐ ๐ท๐จยฒ โ ๐ ๐จ๐ฎยฒ.
p.9
Position, Velocity, and Acceleration
What are the equations for the x and y coordinates of the particle?
x(t) = t^3 - 32t and y(t) = 5t^2 + 12.
What constants are involved in the trajectory equation of a projectile?
๐ฃโ (initial velocity), โ
โ (launch angle), and ๐ (acceleration due to gravity).
What does the horizontal component of velocity equal during projectile motion?
๐ ๐ = ๐ ๐๐ + ๐ ๐ ๐ = ๐ ๐ ๐๐๐ โ
๐.
What is the trajectory equation of a projectile after eliminating time (t)?
๐ฆ = tan โ
โ ๐ฅ โ (๐ / (2(๐ฃโ cos โ
โ)ยฒ)) ๐ฅยฒ
What does the note indicate about the relationship between ground speed and air speed when heading into the wind?
The ground speed is less than the air speed.
p.34
Uniform Circular Motion
In uniform circular motion, what is the direction of the acceleration vector?
Always radially inward toward the center of the circle.
What does the velocity vector represent in relation to the particle's path?
The velocity vector is tangent to the path of the particle at every point.
p.33
Uniform Circular Motion
How is the change in velocity represented in the context of uniform circular motion?
By drawing โv from the midpoint of the arc P1P2.
What is the formula for the vertical component of velocity in projectile motion?
v_y = v_0 sin(ฮธ_0) - g t
p.15
Motion with Constant Acceleration
What is the equation for velocity in the x-direction (๐๐) over time?
๐๐ = ๐๐๐ + ๐๐๐ = 9.0 + 1.7t.
p.15
Motion with Constant Acceleration
What is the equation for position in the y-direction (๐) over time?
๐ = ๐๐ + ๐๐๐๐ + 1/2 ๐๐๐ยฒ = 0 + 0 + 0.27tยฒ.
What is the relationship between the angle โ
and the velocity components?
tan โ
= ๐ ๐ / ๐ ๐.
p.32
Uniform Circular Motion
What does the velocity vector at P1 represent?
It is a vector tangent to the curve at P1, denoted as v1.
p.13
Motion with Constant Acceleration
What is the initial position of a particle at t = 0?
๐โ = ๐โ๐ข + ๐โ๐ฃ + ๐โ๐ค
p.13
Motion with Constant Acceleration
What is the initial velocity of a particle?
๐ = ๐โโ๐ข + ๐แตงโ๐ฃ + ๐๐โ๐ค
p.13
Motion with Constant Acceleration
What is the vector equation for velocity with constant acceleration?
๐ = ๐โ + ๐๐ก
p.15
Motion with Constant Acceleration
What is the equation for velocity in the y-direction (๐๐) over time?
๐๐ = ๐๐๐ + ๐๐๐ = 0 + 0.54t.
What is the resultant velocity of the plane with respect to the ground?
Approximately 230 km/h at an angle northeast.
How is the angle with respect to the ground calculated?
ฮฑ = tanโปยน(๐ ๐จ๐ฎ / ๐ ๐ท๐จ).
p.15
Motion with Constant Acceleration
What is the acceleration in the x-direction (๐๐) for motion with constant acceleration?
๐๐ = ๐๐๐๐ 10ยฐ = 1.7 m/sยฒ.
p.9
Position, Velocity, and Acceleration
What is the acceleration of the particle at t = 3 s?
a_x(3) = 2(3) = 6 m/sยฒ and a_y(3) = 10 m/sยฒ.
p.9
Position, Velocity, and Acceleration
How do you calculate the acceleration of the particle?
Acceleration is the derivative of velocity with respect to time: a(t) = dv/dt.
p.13
Motion with Constant Acceleration
What is the equation relating initial and final velocities with acceleration?
๐ยฒ = ๐โยฒ + 2๐(๐ - ๐โ)
p.13
Motion with Constant Acceleration
What is the equation for position when considering final velocity?
๐ = ๐โ + ๐๐ก - rac{1}{2}๐๐กยฒ
p.15
Motion with Constant Acceleration
What is the equation for position in the x-direction (๐) over time?
๐ = ๐๐ + ๐๐๐๐ + 1/2 ๐๐๐ยฒ = 0 + 9t + 0.85tยฒ.
