#1.A ball is thrown up a hill with initial speed of v0 at an angle θ. The hill inclined at angle of φ.
At what time will the ball land?
Are given: v0, θ, and φ
Time of landing?
The landing point may be described as h=xtanφ and h=(𝑣0 𝑠𝑖𝑛𝜃)𝑡 −
As for horizontal displacement x=(𝑣0 𝑐𝑜𝑠𝜃)𝑡;
𝑔𝑡 2
2
𝑔𝑡2
ℎ (𝑣0 𝑠𝑖𝑛𝜃)𝑡−
𝑔𝑡 2
𝑔𝑡
Then tanφ=𝑥 = (𝑣 𝑐𝑜𝑠𝜃)𝑡 2 =𝑡𝑎𝑛𝜃 -2(𝑣 𝑐𝑜𝑠𝜃)𝑡= 𝑡𝑎𝑛𝜃 − 2(𝑣 𝑐𝑜𝑠𝜃), that gives
0
0
0
(𝑡𝑎𝑛𝜃−𝑡𝑎𝑛φ)
2𝑣0
2𝑣0
t=
2(𝑣0 𝑐𝑜𝑠𝜃) = 𝑔 (𝑡𝑎𝑛𝜃 − 𝑡𝑎𝑛φ)𝑐𝑜𝑠𝜃= 𝑔 (𝑠𝑖𝑛𝜃 - 𝑡𝑎𝑛φ𝑐𝑜𝑠𝜃)
𝑔
⃗ 𝒂 = (200km/h)𝒊̂ + (20 km/h)𝐣̂, where 𝒊̂ points
#2. An aircraft is moving in still air with a constant velocity of 𝒗
⃗ 𝒘 = (20𝒕)𝒊̂ – (30𝒕𝟐 )𝒋̂. The pilot makes
east and 𝐣̂, points north. Suddenly at t=0, the wind gusts with velocity of 𝒗
no attempt to compencate for the wind, what will the plane’s displacement be in 1 h with respect to the ground?
⃗ 𝒂,𝑹 =𝒗
⃗ 𝒂 +𝒗
⃗ 𝒘 = (𝟐𝟎𝟎 + 𝟐𝟎𝒕)𝒊̂ +(20-30𝒕𝟐 )𝒋̂.
The resultant vector is the sum of those two, 𝒗
1
⃗⃗⃗⃗ (t)=∫1 𝒗
⃗ 𝑑𝑡 =[(200t+10𝑡 2 )𝒊̂ +(20t-10𝑡 3 )𝒋̂] =210𝒊̂ +10𝒋̂.
Thus, ∆𝑟
𝑜 𝒂,𝑹
0
⃗ 𝟏 =(𝒊̂ +6𝐣̂) 𝑎𝑛𝑑 𝒗
⃗ 𝟐 =(−2𝒊̂ +2𝐣̂).
#3. Two boats initially next to each other begin moving with velocities 𝒗
What is the rate at which the distance between the boats is increasing?
#4. An object begins accelerating from rest at a constant acceleration a=2i-4j. How far is the object at time t=1?
#5. An object is launched from the ground with a speed v and an angle of elevation θ. Find the difference
between the object’s max and min speeds?
#6. Two balls are simultaneously launched off the top of a cliff with the same initial speeds and angles (see
Fig.) What is the velocity of the top projectile with respect to the bottom projectile as a function of time?
#7. A ball is launched off the cliff of height h, with an initial speed of v, and angle of elevation of θ above the
horizontal. How long is the ball in the air?
Section 3.2 The Acceleration Vector
3.7 .. CALC The
coordinates of a bird flying in the xy-plane are given by x(t) = αt and y(t) = 3.0 m - βt2, where
α =2.4 m/s and β=1.2 m/s2 .
(a) Sketch the path of the bird between t=0 and t=2.0 s.
(b) Calculate the velocity and acceleration vectors of the bird as functions of time.
(c )Calculate the magnitude and direction of the bird’s velocity and acceleration at t=2.0 s.
(d) Sketch the velocity and acceleration vectors at t=2.0 s. At this instant, is the bird speeding up, is it slowing
down, or is its speed instantaneously not changing? Is the bird turning? If so, in what direction?
⃗ = 𝒃𝒕𝟐 𝒊̂ + 𝒄𝒕𝟑 𝐣̂, where b and c are positive constants, when does the velocity vector make an
𝒓
angle of 45.00 with the x- and y-axes?
3.43 .. CALC If
3.45 .. CP CALC. A
small toy airplane is flying in the xy-plane parallel to the ground. In the time interval t=0 to
⃗ = (1.20 m/s2)∙t𝒊̂+ [12.0 m/s –(2.00 m/s2)∙t]𝐣̂ .
t=1.0 s, its velocity as a function of time is given by 𝒗
At what value of t is the velocity of the plane perpendicular to its acceleration?