General Physics I Lecturer: Rashadat Gadmaliyev Lecture 2: Position vectors, Trajectory, Velocity Speed and Acceleration.

Position vector The motion of a particle is completely known if the particle’s position in space is known at all times. A particle’s position is the location of the particle with respect to a chosen reference point that we can consider to be the origin of a coordinate system.

Displacement Given the data in Table, we can easily determine the change in position of then car for various time intervals. The displacement of a particle is defined as its change in position in some time interval. As it moves from an initial position x i to a final position x f, the displacement of the particle is given x f -x i. We use the Greek letter Δ (delta) to denote the change in a quantity. Therefore, we write the displacement, or change in position, of the particle as

Distance It is very important to recognize the difference between displacement and distance traveled. Distance is the length of a path followed by a particle. The SI unit of distance is meter (m).Consider, for example, the basketball players. If a player runs from his own basket down the court to the other team’s basket and then returns to his own basket, the displacement of the player during this time interval is zero, because he ended up at the same point as he started. During this time interval, however, he covered a distance of twice the length of the basketball court.

Average velocity The average velocity v x of a particle is defined as the particle’s displacement ∆x divided by the time interval ∆t during which that displacement occurs: where the subscript x indicates motion along the x axis. From this definition we see that average velocity has dimensions of length divided by time (L/T)—meters per second (m/s) in SI units. The average velocity of a particle moving in one dimension can be positive or negative, depending on the sign of the displacement. If the coordinate of the particle increases in time then ∆x is positive and v x is positive. This case corresponds to a particle moving in the positive x direction.

Average speed The average speed of a particle, a scalar quantity, is defined as the total distance traveled divided by the total time interval required to travel that distance: of the equation are the same. The SI unit of average speed is the same as the unit of average velocity: meters per second (m/s). However, unlike average velocity, average speed has no direction and hence carries no algebraic sign. Notice the distinction between average velocity and average speed—average velocity is the displacement divided by the time interval, while average speed is the distance divided by the time interval.

Example: Find the displacement, average velocity, and average speed of the car in Figure between positions A and F. Solution: From the position–time graph given in Figure note that x A = 30 m at t A = 0 s and that x F =-53 m at t F = 50 s. Using these values along with the definition of displacement, Δ x = x F - x A =-53 m - 30 m =-83 m

The average velocity is If we adopt the assumption that the details of the car’s position are described by the curve in Figure then the distance traveled is 22 m (from A to B) plus 105 m (from B to F) for a total of 127 m. We find the car’s average speed for this trip by dividing the distance by the total time

Instantaneous Velocity and Speed The instantaneous velocity v x equals the limiting value of the ratio Δ x/ Δt as Δ t approaches zero: In calculus notation, this limit is called the derivative of x with respect to t, written dx/dt: The slope of position-time graph at any times gives instantaneous velocity. The instantaneous speed of a particle is defined as the magnitude of its instantaneous velocity.

Example: A particle moves along the x axis. Its position varies with time according to the expression x =-4t + 2t 2 where x is in meters and t is in seconds. (A) Determine the displacement of the particle in the time intervals t = 0 to t = 1 s and t = 1 s to t = 3 s. (B) Calculate the average velocity during these two time in- tervals. (page 30) (B) Calculate the instantaneous velocity at the time i) t=0 s ii)t=1 s and iii)t=3 s Solution:

Acceleration When the velocity of a particle changes with time, the particle is said to be accelerating. For example, the magnitude of the velocity of a car increases when you step on the gas and decreases when you apply the brakes. Let us see how to quantify acceleration. The average acceleration a x of the particle is defined as the change in velocity Δv x divided by the time interval Δt during which that change occurs: acceleration has dimensions of length divided by time squared, or L/T 2. The SI unit of acceleration is meters per second squared (m/s 2 ).

In some situations, the value of the average acceleration may be different over different time intervals. It is therefore useful to define the instantaneous acceleration as the limit of the average acceleration as Δt approaches zero. If we imagine that point A is brought closer and closer to point B in Figure and we take the limit of Δ v x /Δ t as Δ t approaches zero, we obtain the instantaneous acceleration

From now on we shall use the term acceleration to mean instantaneous acceleration. Because v x = dx/dt, the acceleration can also be written That is, in one-dimensional motion, the acceleration equals the second derivative of x with respect to time.

Example: The position of an object moving along the x axis varies with time as in Figure Graph the velocity versus time and the acceleration versus time for the object. Solution:

Example: The velocity of a particle moving along the x axis varies in time according to the expression v x =(40 - 5t 2 ) m/s, where t is in seconds. (A) Find the average acceleration in the time interval t= 0 to t = 2.0 s. (B) Determine the acceleration at t = 2.0 s. (page 33) Solution:

Example: The position of a particle moving along the x axis varies in time according to the expression x =(40+2t - 5t 2 )m, where t is in seconds. (A) Find the average acceleration in the time interval t= 0 to t = 2.0 s. (B) Determine the acceleration at t = 2.0 s. Solution: