Introduction to Mechanics Kinematics Equations

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1 Introduction to Mechanics Kinematics Equations Lana Sheridan De Anza College Jan, 018

2 Last time more practice with graphs introduced the kinematics equations

3 Overview rest of the kinematics equations derivations using the equations

4 The Kinematics Equations For constant acceleration: v = v 0 + at x = v 0 t + 1 at x = vt 1 at x = v 0 + v t v = v 0 + a x For zero acceleration: x = vt

5 The Kinematics Equations: the no-displacement equation From the definition of average acceleration: a avg = v t v = v v 0 and starting at time t = 0 means t = t 0 = t.

6 The Kinematics Equations: the no-displacement equation From the definition of average acceleration: a avg = v t v = v v 0 and starting at time t = 0 means t = t 0 = t. For constant acceleration a avg = a, so a = v v 0 t v(t) = v 0 + at (1) where v 0 is the velocity at t = 0 and v(t) is the velocity at time t.

7 The Kinematics Equations: the no-acceleration equation From the definition of average velocity: v avg = x t and v avg = v 0+v Equating them, and multiplying by t: ( ) v0 + v x = t ()

8 The Kinematics Equations: the no-final-velocity equation Using the equation and the equation replace v in the first equation. ( ) v0 + v x = t v = v 0 + at x = ( v0 + (v 0 + at) ) t For constant acceleration: = v 0 t + 1 at x(t) = x 0 + v 0 t + 1 at (3)

9 Example -6, page 34 A drag racer starts from rest and accelerates at 7.40 m/s. How far has it traveled in (a) 1.00 s, (b).00 s, (c) 3.00 s? 1 Walker Physics, pg 33.

10 Example -6, page 34 tal A drag racer starts from rest and accelerates at 7.40 m/s. How far has it traveled in (a) 1.00 s, (b).00 s, (c) 3.00 s? t 7.40 m/s. How far has it traveled in (a) 1.00 s, (b).00 s, (c) 3.00 s? Sketch: rag racer starts direction. With.40 m/s. Also, elocity is zero, r in the sketch t = 0.00 t = 1.00 s t =.00 s t = 3.00 s ich is constant, and time, we O x and t = 1.00 s: x = x 0 + v 0 t + 1 at = at = 1 at x = m/s s = 3.70 m 11 reduces x = 1 at at t =.00 s: = m/s 1.00 s = 14.8 m = m 1 Walker Physics, pg 33.

11 Example -6, page 34 tal A drag racer starts from rest and accelerates at 7.40 m/s. How far has it traveled in (a) 1.00 s, (b).00 s, (c) 3.00 s? t 7.40 m/s. How far has it traveled in (a) 1.00 s, (b).00 s, (c) 3.00 s? Sketch: rag racer starts direction. With.40 m/s. Also, elocity is zero, r in the sketch t = 0.00 t = 1.00 s t =.00 s t = 3.00 s ich is constant, and time, we Hypothesis: and t = 1.00 s: O x For part (a) the car will have travelled 3.7 m in part (a) because after one second the car will be moving at 7.40 m/s, x = x 0 + v 0 t + 1 at = at = but its average velocity will be less. 1 at x = m/s s = 3.70 m The car will have travelled more than twice as far for part (b) as for part (a). 11 reduces x = 1 at at t =.00 s: = m/s 1.00 s = 14.8 m = m 1 Walker Physics, pg 33. The answer for part (c) will be greater than part (b).

12 Example -6, page 34 A drag racer starts from rest and accelerates at 7.40 m/s. How far has it traveled in (a) 1.00 s, (b).00 s, (c) 3.00 s? Given: a = 7.40 m/s, v 0 = 0 m/s, t. Asked for: x 1 Walker Physics, pg 33.

13 Example -6, page 34 A drag racer starts from rest and accelerates at 7.40 m/s. How far has it traveled in (a) 1.00 s, (b).00 s, (c) 3.00 s? Given: a = 7.40 m/s, v 0 = 0 m/s, t. Asked for: x Strategy: Use equation x = x(t) x 0 = v 0 t + 1 at (a) Letting the x-direction in my sketch be positive: x = 0 v 0 t + 1 at = 1 (7.40 m/s )(1.00 s) = 3.70 m 1 Walker Physics, pg 33.

14 Example -6, page 34 A drag racer starts from rest and accelerates at 7.40 m/s. How far has it traveled in (a) 1.00 s, (b).00 s, (c) 3.00 s? Use the same equation for (b), (c) x = x(t) x 0 = v 0 t + 1 at (b) x = 1 at = 1 (7.40 m/s )(.00 s) = 14.8 m (c) x = 1 at = 1 (7.40 m/s )(3.00 s) = 33.3 m

15 Example -6, page 34 (a) 3.70 m, (b) 14.8 m, (c) 33.3 m Analysis: My hypotheses for (a), (b), and (c) were correct. It makes sense that the distances covered by the car increases with time, and it makes sense that the distance covered in each one second interval is greater than the distance covered in the previous interval since the car is still accelerating. The distance covered over 3 seconds is 9 times the distance covered in 1 second. The car covers 30 m in 3 s, giving an average speed of 10 m/s. We know cars can go much faster than this, so the answer is not unreasonable. 1 Walker Physics, pg 33.

16 The Kinematics Equations: the no-initial-velocity equation We can build a very similar equation to that last one. This time we rearrange v = v 0 + at to give: v 0 = v at And put that into the equation ( ) v0 + v x = t x = For constant acceleration: ( (v at) + v = vt 1 at ) t x(t) = x 0 + vt 1 at (4)

17 The Kinematics Equations: the no-time equation The last equation we will derive is a scalar equation.

18 The Kinematics Equations: the no-time equation The last equation we will derive is a scalar equation. ( ) v0 + v x = t We could also write this as: ( v0 + v ( x) i = ) t i where x, v i, and v f could each be positive or negative.

19 The Kinematics Equations: the no-time equation The last equation we will derive is a scalar equation. ( ) v0 + v x = t We could also write this as: ( v0 + v ( x) i = ) t i where x, v i, and v f could each be positive or negative. We do the same for equation (1): v i = (v 0 + at) i

20 The Kinematics Equations: the no-time equation The last equation we will derive is a scalar equation. ( ) v0 + v x = t We could also write this as: ( v0 + v ( x) = ) t where x, v i, and v f could each be positive or negative. We do the same for equation (1): v = (v 0 + at) Rearranging for t: t = v v 0 a

21 The Kinematics Equations: the no-time equation t = v v 0 a ( v0 + v ; x = ) t Substituting for t in our x equation: x = ( ) ( ) v0 + v v v0 a a x = (v 0 + v)(v v 0 ) so, v = v 0 + a x (5)

22 The Kinematics Equations Summary For constant acceleration: v = v 0 + at x = v 0 t + 1 at x = vt 1 at x = v 0 + v t v = v 0 + a x For zero acceleration: x = vt

23 Summary kinematics equations for constant acceleration Homework graphs worksheet, due tomorrow Jan 3 Walker Physics: Ch, onward from page. Questions: 11, 1, 13; Problems: 49, 55, 57, 65, 67

Kinematics Kinematic Equations and Falling Objects

Kinematics Kinematic Equations and Falling Objects Lana Sheridan De Anza College Sept 28, 2017 Last time kinematic quantities relating graphs Overview derivation of kinematics equations using kinematics