Motion in Two and Three Dimensions

Question 1

True/False:

In projectile motion, the horizontal motion and the vertical motion are dependent of each other; that is, both motion affects the other.

Answer 1

False

Question 2

Derive the equation of the path (trajectory).

Answer 2

y = (tanØ_o)x - gx² / 2(v_ocosØ_o)² .

Question 3

Derive the horizontal range.

Answer 3

R = v_o²sin2Ø_o / g

Question 4

A fly ball is hit to the outfield. During its flight (ignore the effects of the air), what happens to its (a) horizontal and (b) vertical components of velocity? What are the (c) horizontal and (d) vertical components of its acceleration during ascent, during descent, and at the topmost point of its flight?

Answer 4

(a) constant throughout
(b) +v →0 → -v
(c) a_x = 0 throughout.
(d) -g throughout.

Question 5

In uniform circular motion what is constant and what is changing? What is a result to this change?

Answer 5

Speed is constant, velocity is changing only in direction.

Since velocity is changing, the particle is accelerating by definition.

This acceleration is directed radially inward.

Question 6

An object moves at constant speed along a circular path in a horizontal xy plane, with the center at the origin. When the object is at x = -2 m, its velocity is -(4 m/s)j. Give the object’s (a) velocity and (b) acceleration at y = 2 m.

Answer 6

The magnitude of the velocity never changes when dealing with an object that moves at constant speed along a circular path. As well, the velocity vector is tangent to the path, thus with the information given we can deduce that the particle is moving in a clockwise motion.

(a) -(4 m/s)i
(b) a = v²/r radially inward

= (16 m²/s²) / 2 m

= 8 m/s²

Question 7

When two frames of reference A and B are moving relative to each other at constant velocity, the velocity of particle P as measured by an observer in frame A usually differs from that measured from frame B. What are the two measured velocities related by? Derive relative motion in one dimension (Frame B moves past frame A while both observe P).

O

@————-|——>

x_PB

@—O————-|———–>

x_BA x_PA

Answer 7

x_PA = x_BA + x_PB

⇒ v_PA = v_BA + v_PB, taking the time derivative.

⇒ a_PA = a_PB, taking another time derivative.

The accerlation seen by A & B of P are the same only if A & B both move at constant velocity.

Question 8

If a particle travles along a circle or circular arc of radius r at constant speed v, it is said to be in uniform circular motion and has an acceleration a of constant magnitude of ? Also find the period T.

Answer 8

T = 2πr/v

a | = v²/r

Question 9

A ball is to be shot from level ground toward a wall at distance x. Figure 4-43b shows the y component v_y of the ball’s velocity just as it would reach the wall, as a function of that distance x. The scaling is set by v_ys = 5.0 m/s and x_s = 20m. What is the launch angle?

Answer 9

slope of the graph is -1/2 ⇒ Δv_y / Δx = -1/2 = (Δv_y/Δt) / (Δx/Δt) = a_y / v_ox.

⇒ v_0x = 2a_y = 2g = -19.6 m/s.

ø = - tan^-1 (v_y / v_x), since we are below the horizontal line.

Question 10

Answer 10

y = ½a_yt²

vt = ½a_xt²

t =2v/a_x

using sin²Ø = 1 - cos²Ø and envoking the quadratic equation leads to

cosØ = ½

⇒ Ø = 60°

Question 11

Answer 11

t = d / v_ox

Δy = v_oyt - ½at²

NOTE: Since we are below the horizontal axis we will envoke -Ø which will cause sin(-Ø) = -sinØ.

Question 12

Answer 12

(a) t = d/v_ox , Δy = v_oyt - ½at²
(b) v_x = v_ox
(c) v_y = v_oy + at
(d) Highest point ⇒ v_y = 0

so if v_y < 0 then yes otherwise no.

Question 13

Answer 13

Note: At the Δy_max, v_y = 0; on the graph this is where the slope is at the lowest. So, v² = (v_ox² + v_y²)^½ and v_ox = v_x is constant then v_ox = 19 m/s = v_a.

(a) d = v_at where t = 5 sec.
(b) v_o² = (v_ox² + v_oy²)^½

Δy = v_oyt - ½at² where t = 2.5 sec.

Question 14

A soccer ball is kicked from the ground with an initial speed of 19.5 m/s at an upward angle of 45°. A player 55 m away in the direction of the kick starts running to meet the ball at that instant. What must be his average speed if he is to meet the ball just before it hits the ground?

Answer 14

R = 2V02cosøsinø / a , this will be how far the ball will reach.

|R - 55| is the distance the player has to reach.

t = R / Vocosø will give you the time it takes the ball to get there as well as how long the player has to get there.

Finally, Vplayer = |R - 55| / t

Question 15

A cannon located at sea level fires a ball with initial speed 82 m/s and initial angle 45°. The ball lands in the water after traveling a horizontal distance 686 m. How much greater would the horizontal distance have been had the cannon been 30 m higher?

Answer 15

Use the trajectory formula then the quadratic equation for x.

Find the difference between xy=0 and xy=30.