4.0 SPEED VELOCITY AND ACCELERATION

   SPEED

   What's the difference between two identical objects traveling at different speeds? Nearly everyone knows that the one moving faster (the one with the greater speed) will go farther than the one moving slower in the same amount of time.

Either that or they'll tell you that the one moving faster will get where it's going sooner than the slower one. Whatever speed is, it involves both distance and time. "Faster" means either "farther" (greater distance) or "sooner" (less time).

   Doubling one's speed would mean doubling one's distance traveled in a given amount of time. Doubling one's speed would also mean halving the time required to travel a given distance. If you know a little about mathematics, these statements are meaningful and useful. (The symbol v is used for speed because of the association between speed and velocity, which will be discussed shortly.)

   ⇒ Speed is directly proportional to distance when time is constant: v ∝ s (t constant)

   ⇒ Speed is inversely proportional to time when distance is constant: v ∝ 1/t (s constant)

   Combining these two rules together gives the definition of speed in symbolic form.

      v =     s/t

      ☞ This is not the final definition.

   Speed is the rate of change of distance with time. In order to calculate the speed of an object we must know how far it's gone and how long it took to get there. "Farther" and "sooner" correspond to "faster". Let's say you drove a car from New York to Boston. The distance by road is roughly 300 km (200 miles). If the trip takes four hours, what was your speed? Applying the formula above gives…

   v =  s/t ≈ 300 km/4 hour
         = 75 km/h

4.1 VELOCITY

   In order to calculate the speed of an object we need to know how far it's gone and how long it took to get there. A wise person would then ask… What do you mean by how far? Do you want the distance or the displacement?

   Your choice of answer to this question determines what you calculate — speed or velocity.

   ⇒ Average speed is the rate of change of distance with time.

   ⇒ Average velocity is the rate of change of displacement with time.

   ⇒ Instantaneous speed is the first derivative of distance with respect to time.

   ⇒ Instantaneous velocity is the first derivative of displacement with respect to time.

   Speed and velocity are related in much the same way that distance and displacement are related. Speed is a scalar and velocity is a vector. Speed gets the symbol v (italic) and velocity gets the symbol v (boldface). Average values get a bar over the symbol.

   ⇒ average speed v =    Δs/Δt

   ⇒ instantaneous speed a = lim_Δt→0    Δs/Δt = ds/dt

   ⇒ average velocity v =    Δs/Δt

   ⇒ instantaneous velocity v =  lim_Δt→0    Δs/Δt = ds/dt

   Displacement is measured along the shortest path between two points and its magnitude is always less than or equal to the distance. The magnitude of displacement approaches distance as distance approaches zero. That is, distance and displacement are effectively the same (have the same magnitude) when the interval examined is "small". Since speed is based on distance and velocity is based on displacement, these two quantities are effectively the same (have the same magnitude) when the time interval is "small" or, in the language of calculus, the magnitude of an object's average velocity approaches its average speed as the time interval approaches zero.

   Δt → 0     ⇒     v → |v|

   The SI unit of speed and velocity is the meter per second [m/s]


4.2 ACCELERATION

   When the velocity of an object changes it is said to be accelerating. Acceleration is the rate of change of velocity with time.

   In everyday English, the word acceleration is often used to describe a state of increasing speed. For many Americans, their only experience with acceleration comes from car ads. When a commercial shouts "zero to sixty in six point seven seconds" what they're saying here is that this particular car takes 6.7 s to reach a speed of 60 mph starting from a complete stop. This example illustrates acceleration as it is commonly understood, but acceleration in physics is much more than just increasing speed.

   Any change in the velocity of an object results in an acceleration: increasing speed (what people usually mean when they say acceleration), decreasing speed (also called deceleration or retardation), or changing direction (called centripetal acceleration). Yes, that's right, a change in the direction of motion results in an acceleration even if the moving object neither sped up nor slowed down. That's because acceleration depends on the change in velocity and velocity is a vector quantity — one with both magnitude and direction. Thus, a falling apple accelerates, a car stopping at a traffic light accelerates, and an orbiting planet accelerates. Acceleration occurs anytime an object's speed increases or decreases, or it changes direction.

   Much like velocity, there are two kinds of acceleration: average and instantaneous. Average acceleration is determined over a "long" time interval. The word long in this context means finite — something with a beginning and an end. The velocity at the beginning of this interval is called the initial velocity, represented by the symbol v0 (vee nought), and the velocity at the end is called the final velocity, represented by the symbol v (vee). Average acceleration is a quantity calculated from two velocity measurements.

   Summary

    Acceleration is the rate of change of velocity with time.
        As a vector it must be stated with both magnitude and direction.

    Acceleration occurs anytime an object's…
        speed increases
        speed decreases
        direction of motion changes

    Average acceleration…
        is measured over a non-zero time interval
        is represented by the symbol a_ave or a^_(overline)

    Instantaneous acceleration…
        is the limit of average acceleration as the time interval approaches zero
        is the first derivative of velocity with respect to time
        is the second derivative of displacement with respect to time
        is represented by the symbol a (boldface)

    The various forms of acceleration are defined by the following equations…

    average acceleration  a = Δv/Δt
    instantaneous acceleration a = lim_Δt→0 Δv/Δt = dv/dt = d²s/dt²
                  

    The SI unit of acceleration is the meter per second squared (m/s²)

   The standard acceleration due to gravity…
       is a natural unit of acceleration
       is represented by the symbol g (roman)
       is equal to 9.80665 m/s² by definition
       is often rounded to 9.8 m/s² for convenience
       is sometimes called the unit of "g force" even though it is a unit of acceleration

 

 [1] What is Velocity
 [2] Diffrentiate between speed and velocity
 [3] What is acceleration
 [4] Explain what is meant by acceleration of free fall due to gravity
 [5] State 2 reason why g varies on the surface of the earth
 [6] A body fall from rest with an acceleration of 10m/s2, What is the velocity after 5s
 [7] A stone is thrown vertically upward from ground level with a speed of 25m/s, at what time is it 5m above the ground ?
 [8] What is average speed
 [9] A baseball is pitched at 40 m/s (90 mph) in a Major League game. The batter hits the ball on a line drive straight toward the pitcher at 50 m/s (112 mph). Determine the the acceleration of the ball if it was in contact with the bat for 1/30s.
 [10] A particle is projected horizontally at 15ms^-1 from a height of 20m.
Calculate the horizontal distance covered by the particle just before hitting the ground. [g = 10 ms-2]