Exercises in 1D and 2D Motion
1. The position of a particle is given by:
𝒙 = 𝟑. 𝟎𝒕 + 𝟎. 𝟓𝒕𝟑
(a)
Find the instantaneous velocity at 2.0s. (Ans: 9.0m/s)
(b)
Calculate the average velocity between 1.0 s and 3.0 s. (Ans: 9.5m/s)
2. A particle moves along the x-axis. Its position is given by the equation:
𝒙 = 𝟐 + 𝟑𝒕 − 𝟒𝒕𝟐
with 𝑥 in metres and 𝑡 in seconds. Determine:
(a) Its position when it changes direction.
(b) Its velocity when it returns to the position it had at 𝒕 = 0s.
3. The track of a cosmic ray particle in a photographic emulsion is found empirically to be
described by the expression:
𝒓⃗ = (𝟑. 𝟎𝒕𝟑 − 𝟔. 𝟐𝒕) ̂ + (𝟓. 𝟎 − 𝟖. 𝟐𝒕𝟒 ) ̂
Determine the velocity and acceleration in 2.0s.
4. A student kicks a soccer ball, giving it an initial speed of 23.5 m/s at an angle of 33.7°
above the horizontal (=ground).
(a) How high will the ball go?
(b) How far will it travel?
(c) Find its total flight time.
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5. A truck travels at a speed of 64 km/h, heading towards town A. Three hours later, a
jeep travelling at a speed of 96 km/h, begins its journey from the same place. When
does the Jeep catch up with the truck?
6. A golf ball is struck at an angle of θ = 37.5º with an initial speed of 38.4 m/s.
(a) Find the maximum height of the object and the maximum horizontal distance.
(b) Find the minimum speed of the object during the motion.
(c) Find the total time in the air.
7. Ship A is 100 km due west of ship B. Ship A is travelling at 15.0 km/h at 40.0° north
of east. Ship B is travelling at 12 km/h at 53.5° north of west.
(a) Will the two ships collide? How can you tell?
(b) If they do, when will this occur?
8. A car is speeding at a constant 25 m/s in a school zone. A police car starts from rest
just as the speeder passes by it and accelerates at a constant rate of 5.0 m/s2.
(a) When does the police car catch the speeding car?
(b) How fast is the police car travelling when it catches up with the speeder?
9. A particle moves in the xy-plane, starting from the origin at t = 0 with an initial
velocity having an x component of 20 m/s and a y component of -15 m/s. The
particle experiences an acceleration in the x direction, given by ax = 4.0 m/s2.
(a) Determine the total velocity vector at any time.
(b) Calculate the velocity and speed of the particle at t = 5.0 s and the angle the
velocity vector makes with the x axis.
(c) Determine the x and y coordinates of the particle at any time t and its position
vector at this time.
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10.The position of a particle as a function of time is given by:
𝒓⃗ = [(𝟓. 𝟎𝐦/𝐬)𝐭 + (𝟔. 𝟎𝐦/𝒔𝟐 )𝒕𝟐 ] ̂ + [(𝟕. 𝟎𝐦) − (𝟑. 𝟎 𝐦/𝒔𝟑 )𝒕𝟑 ] ̂
where r is in meters and t is in seconds,
(a) What is the particle’s displacement between t1 = 2.0 s and t2 = 3.0 s?
(b) Find the particle’s instantaneous velocity and acceleration as a function of time,
(c) Evaluate 𝒗⃗ and 𝒂⃗ at t = 3.0 s.
11.An object moves along the x-axis according to the equation:
𝒙 = 𝟑. 𝟎𝟎𝒕𝟐 − 𝟐. 𝟎𝟎𝒕 + 𝟑. 𝟎𝟎;
where x is in meters and t is in seconds. Determine
(a) The average velocity between t = 2.00s and t = 3.00s;
(b) The instantaneous velocity at t = 2.00s and at t = 3.00s;
(c) The average acceleration between t = 2.00s and t = 3.00s;
(d) The instantaneous acceleration at t = 2.00s and t = 3.00s.
(e) At what time is the object at rest?
12.A football is kicked at an angle  = 37.0° with a velocity of 20.0 m/s.
Calculate,
(a) The maximum height,
(b) The time of travel before the football hits the ground,
(c) How far away it hits the ground,
(d) The velocity vector at the maximum height,
(e) The acceleration vector at maximum height. Assume the ball leaves the
foot at ground level and ignore air resistance and rotation of the ball.
13. A race car accelerates uniformly from 18.5 m/s to 46.1 m/s in 2.47
seconds. Determine the acceleration of the car and the distance travelled.
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14. A fish swimming in a horizontal plane has velocity 𝒗⃗𝒊 = (𝟒. 𝟎𝟎 ̂ + 𝟏. 𝟎𝟎 ̂) m/s at
a point in the ocean where the position relative to a certain rock is
𝒓⃗𝒊 = (𝟏𝟎. 𝟎 ̂ − 𝟒. 𝟎𝟎 ̂) m. After the fish swims with constant acceleration
for 20.0 s, its velocity is 𝒗⃗𝒇 = (𝟐𝟎. 𝟎 ̂ − 𝟓. 𝟎𝟎 ̂) m/s.
(a) What are the components of the acceleration of the fish?
(b) What is the direction of its acceleration with respect to unit vector ̂?
(c) If the fish maintains constant acceleration, where is it at 𝒕 = 25.0 s and
in what direction is it moving?
15. A car travelling east at a constant speed of 20.0 m.s-1 begins to accelerate at 3.80
m.s-2 when it is 200 m behind a truck currently mobbing east at a constant speed
of 42.0 m.s-1.
(a) How long will the car catch up with the truck?
(b) How far will the truck and car travel during this period?
16. Car X and Car Y are 10 km apart driving towards another. If Car X travels at
30.0m/s and Car Y travels at 45.0 m/s, where will the cars meet relative to Car X?
17. Train 1 and Train 2 are 200 km apart pointing toward one another. Train 1
leaves 2.00 hrs after Train 2. If Train 1 travels at 20.0 km/h east and Train 2
west, where will they meet relative to Train 1.
18. The position of an electron is given by 𝒓⃗ = 𝟑. 𝟎𝒕 ̂ − 𝟒. 𝟎𝒕 𝟐 ̂ + 𝟐. 𝟎𝒌
(where t is in seconds and the coefficients have the proper units for r to be in
meters). (a) What is v(t) for the electron? (b) In unit–vector notation, what is v
at t = 2.0 s? (c) What are the magnitude and direction of v just then?
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