Two bodies with different accelerations

Vectors

Adding vectors using their components

Three players on a reality TV show are brought to the center of a large, flat field. Each is given a meter stick, a compass, a calculator, a shovel, and (in a different order for each contestant) the following three displacements:

: 72.4 m, 32.0° east of north

: 57.3 m, 36.0° south of west

: 17.8 m due south

The three displacements lead to the point in the field where the keys to a new Porsche are buried. Two players start measuring immediately, but the winner first calculates where to go. What does she calculate?

 

Using unit vectors

Given the two displacements find the magnitude of the displacement

= (6.00 + 3.00 – 1.00 ) m and

= (4.00 – 5.00 + 8.00 ) m

 

Finding an angle with the scalar product

Find the angle between the vectors

= (2.00 + 3.00 – 1.00 ) m and

= (– 4.00 + 2.00 – 1.00 ) m

 

Calculating a vector product

Vector has magnitude 6 units and is in the direction of the +x-axis. Vector has magnitude 4 units and lies in the xy-plane, making an angle of 30° with the +x-axis. Find the vector product =

Components of Vectors

Vector is in the direction 34o clockwise from the –y-axis . The x-component of is Ax = - 16.0 m . (a) What is the y-component ? (b) What is the magnitude of ?

 

6. Find the magnitude and direction of the vector represented by the following pairs of components: (a) Ax = - 8.60 cm, Ay = 5.20 cm.

 

7.Given two vectors = 4.00 + 7.00 and = 5.00 – 2.00

(a) find the magnitude of each vector; (b) write an expression for the vector difference using unit vectors.

 

8. Given two vectors = 4.00 + 7.00 and = 5.00 – 2.00

(a) find the magnitude and direction of the vector difference

 

9. Find the scalar product of the two vectors = 4.00 + 7.00 and = 5.00 – 2.00

Find the angle between these two vectors.

 

10. Given two vectors = 4.00 + 7.00 and = 5.00 – 2.00

Find the vector product. What is the magnitude of the vector product?

Kinematics

Average and instantaneous velocities

A cheetah is crouched 20 m to the east of an observer (Fig. 2.6a). At time t = 0 the cheetah begins to run due east toward an antelope that is 50 m to the east of the observer. During the first 2.0 s of the attack, the cheetah’s coordinate x varies with time according to the equation x = 20 m +(5.0 m/s2)t2

(a) Find the cheetah’s displacement between t1 = 1.0 s and t2 = 2.0 s.

(b) Find its average velocity during that interval.

(c) Find its instantaneous velocity at t1 = 1.0 s by taking t = 0.1 s, then 0.01 s, then 0.001 s

(d) Derive an expression for the cheetah’s instantaneous velocity as a function of time, and use it to find ʋx at t =1 s and t = 2 s

 

Constant-acceleration calculations

A motorcyclist heading east through a small town accelerates at a constant 4.00 m/s2 after he leaves the city limits (Fig. 2.20). At t = 0 time he is 5.0 m east of the city-limits signpost, moving east at 15 m/s. (a) Find his position and velocity at t = 2.0 s.(b) Where is he when his velocity is 25 m/s?

 

Two bodies with different accelerations

A motorist traveling with a constant speed of 15 m/s passes a school-crossing corner, where the speed limit is 10 m/s. Just as the motorist passes the school-crossing sign, a police officer on a motorcycle stopped there starts in pursuit with a constant acceleration of 3.0 m/s2

(a) How much time elapses before the officer passes the motorist? (b) What is the officer’s speed at that time? (c) At that time, what distance has each vehicle traveled?

 

A freely falling coin

A one-euro coin is dropped from the Leaning Tower of Pisa and falls freely from rest. What are its position and velocity after 1.0 s, 2.0 s, and 3.0 s?