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PC1431 Physics IE: Tutorial 1

Question 1

A boy stands at the peak of a hill which slopes downwards uniformly at an angle as

shown in Figure1. At what angle from the horizontal should he throw a rock so that

it has the greatest range, R?

v

R

Figure 1: Calculating the angle for maximum distance

[Ans: = /4/2]

Question 2 : Human cannonball

In a circus stunt, a clown was fired out of a cannon to land on a truck. The clown was

fired with speed v0 at an angle of 53.1 above the horizontal. The truck was initially

5.00 m ahead of the clown and moved forward at the same instant that the clown was

fired off, with a constant acceleration ofg/4.

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Figure 2: The Human Cannonball

You may assume that the clown landed at the same level from which he was fired.

Also the truck starts from rest at the moment the clown was fired. At what value ofv0

would the stunt be successful? [Ans: 8.75 m/s]

Question 3

A car moving at 18.1 ms1 enters a curve that describes a quarter turn of radius 120

m from Ato B as shown in Figure3.

B

R

A

Figure 3: Turning a corner

The driver gently applies the brakes, slowing the car down at a constant rate of

0.65 ms2. Just before emerging from the turn at B, what is the angle between the

acceleration vector and the direction of motion?

[Ans: 133]

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Question 4

A small boat is headed for a harbour 32 km north-west of its current position when

it is suddenly engulfed in a heavy fog. The captain maintains a compass bearing of

north-west and a speed of 10 km/hr relative to the water. Three hours later the foglifts and the captain notes that he is now exactly 4.0 km south of the harbour. Taking

to be directed along east and to be directed north, what is the velocity v of the

current during those three hours? (You may assume that v is a constant.)

[Ans: v=(0.47km/hr) (0.86km/hr)]

Question 5 : 2D Kinematics part I

The trajectory of a particle is described by the vector in Cartesian coordinates

r(t) =R (cos t + sin t ) , (1)

where and Rpositive constants. What is the particles corresponding velocity v and

accelerationa? Find also the magnitudes of each vector. Calculate the scalar products

ar anda v.

Question 6 : 2D Kinematics part II (for practice)

Consider now a trajectory

r= R (cos + sin ) , (2)

where is a function oft.

(a) Show the following two results:

|v|= R ddt

, (3)

a=R (cos + sin )

d

dt

2+R ( sin + cos )

d2

dt2

(4)

What are |v|andaif given that = t, where is a constant?

(b) Let us assign the names

a1=R (cos + sin )

d

dt

2

, a2=R ( sin + cos )

d2

dt2

(5)

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such that Eq. (4) is represented asa=a1+ a2. Since we know from the lectures

that totalacceleration is the sum of tangential and centripetal acceleration

a=acent+ atan, (6)

which vectors correspond to a1 and a2? (Hint: remember for uniform circular

motion, the tangential acceleration is zero.)

Question 7 (for practice)

The speed of a particle moving in a circle 2.0 m in radius increases at the constant rate

of 4.4 m/s2. At the instant when the magnitude of the total acceleration is 6.0 m/s2,

what is the speed of the particle?

[Ans: 2.9 m/s]

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