Purdue Prelecture Content IMPACT
- 1. Section 1.7 Velocity Average Speed: d = distance traveled ∆t = tf – ti = time interval to travel distance d This is a scalar quantity, i.e., a number (>0). ρ ρ ρ r ρφ − ρ ∆ρ ι Average Velocity: vave = = τφ − τι ∆τ Average velocity is a vector quantity. It has both a magnitude and direction.
- 2. The gridlines below are 1 meter apart. An object travels at a constant r speed from its initial location specified by the position vector ri (ti ) = 2, 2, 0 µ r to its final location specified by the position vector rf (t f ) = 4, 3, 0 µ It takes 2 seconds to travel from the initial position to the final position. ρ ρ r ρφ − ρ (4 − 2),(3 − 2), 0 r 2 vave = ι = µ / σ= 1, 0.5, 0 µ / σ vave = 12 + ( 0.5 ) µ / σ= 1.1 µ / σ ∆τ 2 ρ −ri r ρ rf − ρι r rf r rf r ri r vave (Click to activate animation)
- 3. In the previous slide, we saw that the magnitude of the average velocity was less than the average speed. In the figure below, we have r displayed the velocity vector v1 for the x-portion of the motion and the r velocity vector v2 for the y-portion of the motion. Each has a magnitude of 2 m/s, the same as the average speed. The figure also shows the r ρ ρ r r vector vsum = ϖ + ϖ Is vsum equal to vave ? 1 2 r vsum r v2 r v1 A) Yes, it is because adding the velocity vectors of the motion gives the net velocity and that is the same as the average velocity. B) No, it is not because the object spends only half of the total time r traveling in each direction, so vsum needs to be half the length shown.
- 4. r r If we know an object’s position ri at time t1, we can predict its position r2 at time t2 given its average velocity during the interval. r ρ ρ ρ ρ rf = ρ + ϖ ε (τφ − τι ) = ρ + ϖ ε ∆τ ι αϖ ι αϖ We will refer to the above equation as the position update formula. This is a vector equation. In terms of Cartesian coordinates, this vector equation is a concise expression of three scalar equations: x f = ξι + ϖ ε, ξ∆τ αϖ ψφ = ψ + ϖ ε, ψ∆τ ι αϖ ζφ = ζι + ϖ ε, ζ∆τ αϖ
- 5. The figure below shows the trajectory of a ball through the air. The points labeled A through F indicate the location of the ball at successive times, one second apart. The curved trajectory is an indication that the ball is interacting with its surroundings. We can calculate the average velocity over three different time intervals as indicated by the three different displacement vectors. Suppose we know the average velocity between location B and C. Can we use this to accurately calculate the position two seconds later? No we cannot. To accurately calculate the position 2 seconds later, we need to account for the object’s interaction with its surroundings. We will address this issue in the near future.
- 6. The three average velocities calculated from the three displacement vectors in figure 1.33 are all different as shown below. Notice that each average velocity vector points from the initial position (B) towards the final position (C, D, E) and that the length of each vector is different. r What is the velocity of the ball at location B? This is v(t B ), the instantaneous velocity. As we shorten the interval between position B and subsequent locations, the calculated velocity vector approaches the tangent to the curve at B. ρ ρ r ∆ρ δρ Instantaneous velocity v=λιµ = δτ→0 ∆τ δτ
- 7. The figure below shows a comet in an elliptical orbit around a star. The total length of the orbit is d meters and it takes ∆t seconds for the comet to complete an orbit. What is the average speed for a complete orbit? C)The comet returns to its initial location, so that the net distance traveled is zero. Therefore d/∆t = 0. D)The average speed cannot be determined because the speed varies as the comet orbits the star. E)The average speed is the total distance traveled divided by the time interval: d/∆t.
- 8. The figure below shows a comet in an elliptical orbit around a star. The total length of the orbit is d meters and it takes ∆t seconds for the comet to complete an orbit. What is the average velocity for a complete orbit? C)The average velocity is zero because the comet returns to its initial position after a complete orbit. D)The average velocity is only well defined for a linear trajectory. E)The average velocity for a complete orbit is different for each starting position, 1, 2, 3, 4, 5.
Editor's Notes
- #2: In this section, we will discuss the difference between the average speed of an object that travels a total distance d in a time interval delta T and the average velocity of the object. The average speed is a scalar quantity and is a number greater than or equal to zero. The average velocity is a vector quantity. It has both a magnitude and a direction.
- #3: The object travels a total distance of 4 meters in 2 seconds. Its average speed is therefore 2 meters per second. We can find the average velocity by finding the displacement vector r_final minus r_iniitial and dividing by the time interval of 2 seconds. What is the average speed in traveling from ri to rf? What is the average velocity in traveling from ri to rf?