Name ________________________
HW -Kinematics with Integrations #2
1. A particle moves along a straight line with its acceleration given by 𝑎 = 12𝑡. At 𝑡 = 0,
the particle is at rest at the origin. What is an expression for the particle’s displacement
in terms of time?
2. The velocity v of a particle at time t is given by v = e−2t + 12t. The displacement of the
particle at time t is s. Given that s = 2 when t = 0, express s in terms of t.
3. A particle moves along a straight line with its position given by
𝑠 = 𝑡 3 − 9𝑡 2 + 24𝑡 − 16. At what time(s) does it change direction?
4. A particle moves along a straight line with is velocity given by 𝑣 = 4𝑡 − 𝑡 2 + 15. What
is the average velocity of the particle during 𝑡 = 0 to 𝑡 = 3?
5. A particle moves along a straight line with its velocity given by 𝑣 = 𝑒 𝑡 − 𝑡. What is the
total distance traveled by the particle from 𝑡 = 0 to 𝑡 = 1?
6. A particle moves along a straight line with its velocity given by 𝑣 = 𝑒 𝑡 + 4𝑡. What is the
average velocity of the particle during 𝑡 = 0 to 𝑡 = 2 (leave your answer in terms of e)?
7. The acceleration of a car traveling on a straight track along the y-axis is given by the
equation 𝑎 = 5, where a is in meters per second squared and t is in seconds. If at 𝑡 = 0
the car’s velocity is 3 m/s, what is its velocity at 𝑡 = 2?
Name __________________________
Extra credit
In this question, s represents displacement in metres, and t represents time in seconds.
ds
(a)
The velocity v m s–1 of a moving body may be written as v =
= 30 – at, where a is a
dt
constant. Given that s = 0 when t = 0, find an expression for s in terms of a and t.
Trains approaching a station start to slow down when they pass a signal which is 200 m from the
station.
(b)
The velocity of Train 1 t seconds after passing the signal is given by v = 30 – 5t.
(i)
Write down its velocity as it passes the signal.
(ii)
(c)
Show that it will stop before reaching the station.
Train 2 slows down so that it stops at the station. Its velocity is given by
v=
ds
= 30 – at, where a is a constant.
dt
(i)
Find, in terms of a, the time taken to stop.
(ii)
Use your solutions to parts (a) and (c)(i) to find the value of a.