Motion with constant acceleration: Which of the following is an example of an object whose velocity is changing, and yet moving with constant acceleration? a) A car driving on the freeway b) A child on a roundabout traveling around in circles c) An apple falling from a tree branch to the ground d) A box sitting on a table e) The Earth orbiting the Sun

Which answers are correct? The strictly correct answer is C. a) Correct, but note that it is a special case. Cars never move with constant acceleration for any sustained period of time, unless that acceleration is zero. Even a slow acceleration of 1 mph/second would increase your speed by 60 mph in a minute, and by a dangerous 120 mph in 2 minutes. Therefore a car could only sustain constant acceleration motion for a long time if the rate of acceleration was very small. This is essentially the case of motion with constant velocity. It is this kind of motion which is typical of freeway driving. But since motion with constant velocity is motion with acceleration = 0 m/s/s, this is a very special case of motion with constant acceleration. b) As we shall see later, motion in a circle typically involves something that is like constant acceleration, but not quite. The amount of acceleration is always the same, but the direction of the acceleration (the “push”) is always changing. Specifically it is always pointed towards the center of the roundabout. c) Falling freely under gravity is the classic example of motion with constant acceleration d) Correct, but another special case. The box, which is not moving at all, has velocity = 0 the whole time. This is a special case of motion with constant velocity, and therefore a special case of a special case of motion with constant acceleration (i.e. still acceleration = 0). e) This case is essentially the same as b, circular motion.

Let's suppose we time the falling apple. We find out that it takes t = 1 s for the apple to fall from the tree branch from the ground. We would like to use this information to calculate the speed v of the apple when it hits the ground, and the height h of the tree branch. We obviously need to know the acceleration of the apple due to gravity, which is easy to remember... The rate of acceleration for all falling objects on the surface of the earth = m/s/s. Why do we write this acceleration as negative? If we want to define motion upwards as the direction of positive velocity and acceleration, then we have to define gravity as a negative acceleration, because it always pushes downwards. We will usually denote the quantity 9.8 m/s/s by the symbol g, so the acceleration at the surface of the Earth is = - g = m/s/s. We have some more information hidden inside our question. If we take the ground as our origin, so that the final position is x=0, then what is the initial position x 0 ? What is the initial velocity of the apple v 0 ? a) x 0 = h; v 0 = 0 b) v 0 = 0; v 0 = v c) v 0 = 0; v 0 = 0 d) v 0 = - h; v 0 = g

Correct answer : a) If we define the ground as the origin, then x=0 (zero displacement) is at the ground. We have also defined upwards to be our direction of positive displacement, and since the branch is at height h above the ground, it follows that the initial position of the apple is x 0 = + h. It is obvious that the apple was not moving when it was attached to the branch, so its initial velocity must have been v 0 = 0.

We now have the following information about our problem. The time of fall is t = 1 s, the acceleration is a = g = m/s/s, the initial velocity is v 0 = 0 m/, the initial position is x 0 = h, and the final position is x = 0 m. We want to find the final velocity v and the height h, using our equations x = x 0 + v 0 t + ½ a t2 and v = v 0 + a t Which of the following is correct a) h = 0 m ; v = 9.8 m/s/s b) h = 9.8/2 m = 4.9 m ; v = m/s/s c) h = 9.8 m ; v = 9.8/2 = m/s/s d) h = 9.8 m ; v = 0 m/s

Correct Answer – B We can rule out answers A and D right away. Answer A would have the height fallen by 0m, in which case the final velocity would have to be 0 m also (also answer A’s final velocity is positive, so the apple is falling upwards!). Answer D has the final velocity be 0 m/s, while the height is 9.8 m, which is also impossible. Looking carefully at B and C, we note that h is found from the first formula x = 0 = h + 0 – ½ g t 2 and so h = ½ g t 2 = ½ (9.8) x 1 = 4.9 m v = 0 – g t = x 1 = m/s/s Which is answer B (the best clue is which formula has the factor of ½ before the acceleration).