Kinematics in Two Dimensions
Free Response Problems
1. A student moves a marker over the surface of a smart board. The position of the marker is given
𝑚
in terms of unit vectors 𝑟⃗ = [0.6 𝑚 + (0.3 𝑠2 ) 𝑡 2 ] 𝑖̂ + (0.5 m/s)t𝑗̂.
a. Find the magnitude and direction of the average velocity of the marker between t = 0
and t = 3 s.
b. Find the magnitude and direction of the instantaneous velocity of the marker at t =0, t =
3 s, t = 5 s.
c. Find the magnitude and the direction of the instantaneous acceleration of the marker at
t = 5 s. Does the acceleration change with time?
d. Sketch the marker’s trajectory from t = 0 to t = 5 s.
2. The displacement of a moving object is presented by the formula 𝑟⃗ = αt2𝑖̂ + βt3𝑗̂, where α =0.8
m/s2, β = 0.01 m/s3.
a. Find the magnitude and direction of the average velocity between t = 0 and t = 4 s.
b. Find the magnitude and direction of the instantaneous velocity at t = 2 s, t = 5 s, t = 7 s.
c. Find the magnitude and direction of the instantaneous acceleration at t = 1 s, t = 3 s, t =
5 s.
d. What is the angle between two vectors – velocity and acceleration at t = 5 s.
3. A toy boat moves on the surface of a pond. The position as a function of time is given by the
following formulas: x(t) = 2.6 (m/s) t and y(t) = 5 m – 1.4 (m/s2)t2.
a. Sketch the path of the toy between t = 0 and t = 4 s.
b. Calculate the velocity and acceleration of the toy as functions of time.
c. Calculate the magnitude and direction of the toy’s velocity and acceleration at t
= 3 s.
d. What is the angle between two vectors – velocity and acceleration at t =3 s.
4. A group of physics students performs an experiment with a small rocket. The rocket’s
acceleration has components ax = (2.4 m/s4)t2 and ay = 8 m/s2 – (1.5 m/s3)t.
a. Calculate the velocity as a function of time.
b. Calculate the position as a function of time.
c. Calculate the maximum height reached by the rocket.
d. Calculate the horizontal displacement of the rocket when it returns to y = 0.
5. A boy dives off a cliff with a running horizontal velocity of 8.6 m/s. The distance from the edge
of the cliff to the surface of the lake is 12 m.
a. How much time will it take the boy to fall from the edge of the cliff to the
surface of water?
b. How far from the cliff will he strike the surface of water?
c. What is the landing velocity?
d. Graph the horizontal and vertical components of the velocity.
6. An air plane flies at a constant horizontal speed of 170 m/s. When the air plane is 1150 m above
the ground level a pilot drops a package.
a. How long it will take the package to reach the ground?
b. How far horizontally will the package fly until it strikes the ground?
c. What is the velocity of the package just before it strikes the ground?
d. How would you compare the velocity of the package and the velocity of an
object dropped from the same height 1150 m.
7. A projectile is fired from the ground with the initial velocity v0 = 45 m/s, 53 å bove the horizontal.
a. How long will it take the projectile to return back to the ground level?
b. What is the maximum horizontal range?
c. What is the maximum height the projectile reaches?
d. What is the projectile’s velocity at the highest point?
e. Which angle will give the projectile the same horizontal range? Does the
projectile reach the same height at this angle?
8. A cannon ball is fired from a cannon located at the edge of 34-m–tall cliff. The initial velocity of
the cannon ball has a magnitude of 540 m/s and an angle 42 ̊above the horizontal.
a. How much time is required for the cannon ball to reach the ground?
b. How far from the cliff will the cannon ball strike the ground?
c. What is the maximum height reached by the cannon ball?
d. What is the landing velocity of the cannon ball (magnitude and direction)?
e. Draw x(t), y(t), vx(t),vy(t), ax(t), ay(t) for the motion of the cannon ball.
9. A fire fighter is trying to shoot water straight to the window located at the second floor of a
house 6 m above the ground. The distance between the fire fighter and the house is 8 m and he
holds the fire hose 1.8 m above the ground. The water leaves the hose with a constant speed of
12.5 m/s. Initially, the fire fighter aims the hose at 53 å bove the horizontal and misses the
window. (we can assume that the hose and the window are in the same vertical plane)
a. How much time it will take for the water flow to reach the house?
b. How far above the window does the water go?
c. What is the velocity of water when it strikes the house?
d. What must be the minimum angle and speed of the flow in order to get water
right into the window?
10. A military jet flies horizontally with a velocity of 450 m/s and 20,000 m above the ground. When
the jet is straight above an artillery gun a shell is fired. Assuming the shell hits the jet.
a. Calculate the horizontal component of the initial velocity of the shell.
b. Calculate the vertical component of the initial velocity of the shell.
c. Calculate the magnitude of the initial velocity of the shell.
d. Calculate the angle of the initial velocity above the horizontal.
11. A ball falls on the inclined surface from the distance H. The inclined surface makes 30˚ with the
horizontal. The ball strikes the surface elastically. Assuming the surface is long enough that the
ball can fall on it for the second time.
a. Find the distance along the inclined surface between the first and second
collisions of the ball with the inclined.
b. If the angle is increased to 45 ˚, what is the new distance between the first and
the second collisions of the ball with the inclined?
12. A basketball is thrown at 60 ˚ below the horizontal toward the floor with an initial velocity of 9
m/s from the distance of 2 m above the floor. Assuming the collision is elastic find the distance
between two consecutive collisions of the ball with the floor.
Answer Key:
1. A. 𝑣𝑎𝑣 = 1.01 m/s, 𝜃 = 29.1˚
B. 𝑣0 = 0.5 m/s & 𝜃 = 90˚, 𝑣3 = 1.87 m/s & 𝜃 = 15.51°, 𝑣5 = 3.04 m/s & 𝜃 = 9.46°
C. 𝑎5 = 0.6 𝑚/𝑠 2 in +x direction
x
D.
t
2. A. 𝑣𝑎𝑣 = 2.05 m/s, 𝜃 = 2.9°
B. 𝑣2 = 3.2 m/s & 𝜃 = 2.15°, 𝑣5 = 8.04 m/s & 𝜃 = 5.36°, 𝑣7 = 11.3 m/s & 𝜃 = 10.62°
C. 𝑎2 = 1.6 𝑚/𝑠 2 & 𝜃 = 2.15°, 𝑎5 = 1.61 𝑚/𝑠 2 & 𝜃 = 6.42°, 𝑎7 = 1.63 𝑚/𝑠 2 & 𝜃 = 10.62°
D. 5.26˚
x
3. A.
t
B. v = 2.6𝑖̂ − 2.8𝑡𝑗̂, a = −2.8𝑗̂
C. v = 8.8 m/s & 𝜃 = 72.8°, a = -2.8 𝑚/𝑠 2 & 𝜃 = 270°
D. 197.2°
4. A. v = 0.8𝑡 3 𝑖̂ + (8𝑡 − 0.75𝑡 2 )𝑗̂
B. x = 0.2𝑡 4 𝑖̂ + (4𝑡 2 − 0.25𝑡 3 )𝑗̂
C. 426.93 m
D. 13107.2 m
5. A. 1.56 s
B. 13.42 m
C. 17.55 m/s
𝑣𝑥
D.
t
t
𝑣𝑦
6. A. 15.32s
B. 2604.4m
C. 226.81m/s
D. The package has greater velocity.
7. A. 7.33s
B. 198.51m
C. 65.9m
D. 45m/s
E. 37˚, No
8. A. 72.3s
B. 29615.81m
C. 6695.21m
D. 530.66m/s, -40.87˚
E.
x
𝑣𝑥
y
𝑣𝑦
𝑎𝑥
𝑎𝑦
t
t
9. A. 1.1s
B. 1.3m
C. 7.5m/s
D. 12.5m/s, 46.4˚
10. A. 450m/s
B. 626.1m/s
C. 771.1m/s
D. 54.3˚
11. A. 4H
B. 4H√2
12. 9.14m
t
t
t
t