19. Exercises
From Mechanics
| Theory | Exercises |
Exercise 19.1
The acceleration, \displaystyle a \displaystyle \text{m}{{\text{s}}^{-2}}, of the cyclist, at time \displaystyle t seconds is given by: \displaystyle a=1-\frac{t}{15} for \displaystyle 0\le t\le 15.
The cyclist travels along a straight road.
a) Given that the cyclist starts at rest, find an expression for the velocity cyclist at time \displaystyle t.
b) What is the speed of the cyclist after 15 seconds?
c) How far does the cyclist travel in the 15 seconds?
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Exercise 19.2
The acceleration, \displaystyle a \displaystyle \text{m}{{\text{s}}^{-2}}, at time \displaystyle t seconds of a particle, which moves along a straight line, is given by:
\displaystyle a=\frac{t}{25} .
a) Given that the initial velocity of the particle is 2 \displaystyle \text{m}{{\text{s}}^{-1}}, find an expression for the velocity of the particle.
b) Find the distance that the particle travels in the first 50 seconds of its motion.
Exercise 19.3
A dragster starts at rest and experiences an acceleration that decreases uniformly from 8 \displaystyle \text{m}{{\text{s}}^{-2}} to zero over a period of 10 seconds. It moves along a straight race track.
a) Show that at time \displaystyle t seconds the acceleration, \displaystyle a \displaystyle \text{m}{{\text{s}}^{-2}} is given by:
\displaystyle a=8-\frac{4t}{5}.
b) Find the speed of the dragster when it stops accelerating at the end of the 10 seconds.
c) Find the distance that the dragster travels in the 10 seconds.
Exercise 19.4
A car, of mass 1000 kg, accelerates for 20 seconds along a straight road. The graph below shows how the resultant force on the car varies with time as it accelerates.
a) Show that the acceleration, \displaystyle a \displaystyle \text{m}{{\text{s}}^{-2}}, at time \displaystyle t seconds is given by
\displaystyle a=\frac{2t}{5} for \displaystyle 0\le t\le 20.
b) Given that the velocity of the car is 8 \displaystyle \text{m}{{\text{s}}^{-1}}
when it starts to accelerate, find an expression for the velocity of the car at time \displaystyle t
seconds.
c) Find the distance travelled by the car in the 20 second period.
Exercise 19.5
A van slows down from a speed of 20 \displaystyle \text{m}{{\text{s}}^{-1}} to 5 \displaystyle \text{m}{{\text{s}}^{-1}} in 40 seconds. At time \displaystyle t seconds the acceleration, \displaystyle a \displaystyle \text{m}{{\text{s}}^{-2}} is given by \displaystyle a=-kt, where \displaystyle k is a constant.
a) Find \displaystyle k.
b) Find the distance that the car travels during the 40 seconds.
Exercise 19.6
A car is moving along a straight road. The velocity of the car at time \displaystyle t seconds is \displaystyle \left( 3{{t}^{2}}+2t+5 \right) \displaystyle \text{m}{{\text{s}}^{-2}}.
a) Find an expression for the acceleration of the car at time \displaystyle t.
b) Find the total distance travelled by the car when \displaystyle t=\text{ 1}0
Exercise 19.7
A particle has mass 4 kg. It moves so that the resultant force, \displaystyle \mathbf{F} N, acting on it at time \displaystyle t seconds is given by:
\displaystyle \mathbf{F}=12t\mathbf{i}+(6-5t)\mathbf{j}
The unit vectors \displaystyle \mathbf{i} and \displaystyle \mathbf{j} are perpendicular.
a) Find the acceleration of the particle.
b) Find the velocity of the particle at time \displaystyle t if it has an initial velocity of -8\displaystyle \mathbf{i} \displaystyle \text{m}{{\text{s}}^{-1}}.
c) Given that the initial position of the particle is (\displaystyle 90\mathbf{i}+20\mathbf{j}) metres, find an expression for the position vector of the particle at time \displaystyle t.
d) Find the distance of the particle from the origin when \displaystyle t=\text{ 2}0.
Exercise 19.8
An aeroplane moves so that at time \displaystyle t seconds its acceleration, \displaystyle \mathbf{a} \displaystyle \text{m}{{\text{s}}^{-2}} is given by:
\displaystyle \mathbf{a}=-2\mathbf{j}
When \displaystyle t=\text{ }0 the velocity of the aeroplane is (40\displaystyle \mathbf{i}) \displaystyle \text{m}{{\text{s}}^{-1}}and its position vector is (400\displaystyle \mathbf{j}) metres. The unit vectors \displaystyle \mathbf{i} and \displaystyle \mathbf{j} are directed east and north respectively.
a) Find the velocity of the aeroplane at time \displaystyle t.
b) Find the position vector of the particle at time \displaystyle t.
c) Find the time when the aeroplane is travelling south east.
d) Find the time when the aeroplane is due east of the origin.
