Examples 2

MATH0370: Introduction to Applied Mathematics 2, 2010–11
Examples 2: More Kinematics
Professor A.M. Rucklidge, 8.17f, Department of Applied Mathematics.
Course web page: http://www.maths.leeds.ac.uk/~alastair/MATH0370/.
Section 1 will be covered in the example class on Friday 11th February. Hand in your
answers to questions from section 2 at the start of the lecture on Monday 14th
February. Throughout, take the gravitational acceleration at the surface of the Earth
as g = 10 m/s 2 .
Section 1: to be attempted in the examples classes
1. (a) A ball is thrown upwards in such a way that it reaches a height of 5 m. At what
upwards speed was it thrown? How long did it take to get to its maximum height?
At what speed should it be thrown in order for it to go twice as high? (Take the
gravitational acceleration on Earth to be g = 10 m/s 2 .)
(b) A space traveller reaches the surface of the newly discovered planet Zorg. She
drops a lead ball from a height of 2 m and finds that it takes 0.5 s for the ball to hit
the ground. What is the gravitational acceleration on Zorg?
2. A rabbit and tortoise are having a race: they both leave the starting position at
time t = 0. The rabbit moves with a constant speed of 9 m/s, while the tortoise
starts from rest but has a constant acceleration of 0.002 m/s 2 . How long (in hours)
does it take for the tortoise to catch the rabbit, and how fast is the tortoise moving
when it does so?
3. (a) Suppose that the speed of a particle moving in one dimension is v = 4s, where
s(t) is the distance it has moved. Find the acceleration of the particle, as a function
of s.
(b) Suppose now that the position of the particle is s(t) = e 4t . Show that, in this
case, the speed and acceleration of the particle are consistent with the equations in
part (a).
4. The lift in a tall building starts at the bottom, accelerates upwards at 2 m/s 2 for
4 s, then decelerates at 1 m/s 2 until it comes to rest at the top floor. How far has
it gone?
5. A stone is dropped from the top of a cliff. One second later, a second stone is thrown
downwards from the same point at 11 m/s. The two stones reach the bottom of the
cliff at the same time. Find the height of the cliff. How fast are the two stones
moving when they hit the bottom of the cliff?
6. A particle moves such that its acceleration is
a(t) = e t i + 6t j + 12t 2 k.
Find the velocity and position of the particle, given that it started at t = 0 from
rest at the origin.

Section 2: to be handed in
1. A ball is thrown vertically upwards with an initial velocity of 9 m/s. How long does
it take to reach its maximum height, and how high does it go? How fast is it moving
when it returns to the ground, and how long does this take? (This experiment was
carried out on the planet Zarquon, where g = 6 m/s 2 .)
2. A car is travelling at 50 m/s along the M1 motorway. How fast is this in km/h?
The car overtakes a stationary police car, which immediately starts to move with
a constant acceleration to chase the speeding vehicle. How fast is the police car
moving when it overtakes the other car? What is the average speed of the police
car during the chase?
3. Suppose that the speed of a particle moving in one dimension is v = 4 √ s, where
s(t) is the distance it has moved. Show that the particle is moving with constant
acceleration, and find the value of this acceleration. If the particle starts from rest
at t = 0, find the speed and distance as a function of time.
4. A train leaves the station and accelerates at 3 m/s 2 for 12 s. How quickly is it
travelling at the end of this period? The train continues at a constant speed for
45 s, and then brakes at a constant deceleration of 2 m/s 2 until it comes to rest.
How long does this take? How far has the train travelled in total? What is the
train’s average speed?
5. A car is travelling at 100 m/s, and is 210 m away from a level crossing, when the
driver notices an oncoming train. He instantly slams on the brakes, which decelerates
the car at a rate of 20 m/s 2 . Does the car cross the tracks, and if so, when?
How long does it take for the car to come to rest?
6. A particle moves such that its acceleration is
a(t) = −8 cos 2t i + 6 j + 12t k.
Find the velocity and position of the particle, given that it started at t = 0 at the
origin with velocity i − 3 j.