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Derivative Examples 2 x (m) 10 Example 3 8 6 Complete the velocity vs. time and acceleration vs. time graphs for an object whose motion results in the position vs. time graph below. 4 2 0 2 4 6 8 2 4 6 8 2 4 6 8 t (s ) -2 -4 v slope of x vs. t graph x v t 4s t 6 s 8m v 2s v 4m s v (m ) s 4 2 0 t (s ) -2 -4 a (m s2 ) 4 0 -4 t (s ) x (m) Derivative Examples 2 10 Example 3 8 6 Complete the velocity vs. time and acceleration vs. time graphs for an object whose motion results in the position vs. time graph below. 4 2 0 2 4 6 8 2 4 6 8 2 4 6 8 t (s ) -2 -4 a slope of v vs. t graph v v a a t t a 4m s 2s a 2m s2 a v (m ) s 4 2 0 4m -2 s 1s a 4 m t (s ) -4 a (m s2 ) 4 s2 0 -4 t (s ) x (m) Derivative Examples 2 10 Example 3 8 6 Complete the velocity vs. time and acceleration vs. time graphs for an object whose motion results in the position vs. time graph below. 4 2 0 4 6 8 2 4 6 8 2 4 6 8 t (s ) -2 v slope of x vs. t graph -4 v (m ) s dx v dt 4 2 0 a slope of v vs. t graph dv a dt 2 d x a 2 dt 2 t (s ) -2 -4 a (m s2 ) 4 The second derivative of the position with respect to time 0 -4 t (s ) Derivative Examples 2 x (m) 10 Example 3 8 6 Complete the velocity vs. time and acceleration vs. time graphs for an object whose motion results in the position vs. time graph below. v slope of x vs. t graph x v The slope of a line connecting t any two points on a x vs. t graph. 4 2 0 2 4 6 8 2 4 6 8 2 4 6 8 t (s ) -2 -4 v (m ) s 4 2 0 t (s ) -2 a slope of v vs. t graph v a The slope of a line connecting t any two points on a v vs. t graph. -4 a (m s2 ) 4 0 -4 t (s ) Derivative Examples 2 Example 4 An object moves with position given by xt 2.0t 3 3.0 calculate the object's a. Instantane ous velocity at t 1.0 s and t 2.0 s. dx v dt d 2.0t 3 3.0 v dt d 2.0t 3 d 3.0 v dt dt v 6.0t 2 2 2 v 1.0 s 6.01.0 v 2.0 s 6.02.0 m m v 1.0 s 6.0 v 2 . 0 s 24 s s Derivative Examples 2 Example 4 continued An object moves with position given by xt 2.0t 3 3.0 calculate the object's b. Instantane ous accelerati on at t 1.0 s and t 2.0 s. dv a dt d 6.0t 2 a dt a 12t a1.0 s 121.0 a 1.0 s 12 m 2 s a2.0 s 122.0 a 2.0 s 24 m 2 s Derivative Examples 2 Example 4 continued An object moves with position given by xt 2.0t 3 3.0 calculate the object's c. Average velocity and average accelerati on from t 1.0 s and t 2.0 s. x v t x 2.0 s x 1.0 s v 2.0 s 1.0 s 2.02.03 3.0 2.01.03 3.0 v 1.0 s v 14 m s Part (a) Notice: Not the simple average of (a) since the velocity function is nonlinear and the rate of change of the velocity is non-constant. Derivative Examples 2 Example 4 continued An object moves with position given by xt 2.0t 3 3.0 calculate the object's c. Average velocity and average accelerati on from t 1.0 s and t 2.0 s. v a t v 2.0 s v 1.0 s a 2.0 s 1.0 s 24 m s 6.0 m s a 1.0 s a 18 m 2 s Part (b) Notice: This is the simple average of (b) since the acceleration function is linear so that the rate of change of the acceleration is constant.