Physics 1110: Mechanics Announcements: CAPA set available in bins. Lectures can be found at the Course Calendar link. Written homework #1 (on website) due at beginning of recitation. The Moving Man simulation is helpful in understanding the relationship between position, velocity, and acceleration. You can find it from the PHET simulations link off the main web page or directly here: phet.colorado.edu/en/simulation/moving-man Web page: http://www.colorado.edu/physics/phys1110/ 1
Velocity versus time graph The velocity can be found from the position vs time slope. The acceleration can be found from the slope of the velocity vs time graph: Δ Steeper slopes correspond to greater acceleration. a Δv (m/s) 4 3 1 1 run: 3 rise: Δ 4 t (s) Remember: knowing if acceleration is positive or negative does not tell you if it is slowing down or speeding up. a! a! v! v! If and are in opposite directions, it is slowing down If and are in the same direction, it is speeding up
Position, velocity, acceleration vs time Constant velocity (m) 100 (m) 100 Constant acceleration v(m/s) 10 v(m/s) 0 a(m/s ) a(m/s ) 3
Clicker question 1 Set frequency to BA Q. A train moves along a straight track. The graph shows the train position as a function of time. The graph shows that the train: position A. speeds up the whole time B. slows down the whole time C. moves at constant velocity D. speeds up part of the time and slows down part of the time E. none of the above The slope is largest at t 0 and continually decreases as time increases so the speed is decreasing (but positive). time 4
Clicker question Q. An object s velocity vs time graph is shown on the right. What best describes the car s acceleration vs time? a a A B Set frequency to BA v t t t a C a D t t E. None of these 5
v(m/s) Relation between velocity & acceleration Eamples of constant acceleration v(m/s) v(m/s) 0 t(s) 0 t(s) 0 t(s) Velocity starts at 0 and increases. Acceleration > 0 Velocity starts at 0 and decreases. Acceleration < 0 Velocity starts positive and decreases to 0 and then decreases more. Acceleration < 0 Acceleration is the change in velocity, not the velocity 6
For this situation: Derivatives If the velocity is not constant and you know the position versus time, how do you find the velocity at a given time? (m) 100 Measure slope of tangent line ( 1m/s ) t Instantaneous velocity: The instantaneous velocity at time t is: v d dt d dt ( ) t ( 1m/s m/s )t Taking the derivative gives you the slope v v(m/s) 0 This matches the graph we drew: lim Δ t 0 Δ d dt 7
Acceleration We define acceleration as change of velocity over time! a avg Δ! v rearranging:! v v! 0 t t 0 so a avg, Δ v 0 t t 0 in 1D v 0 + a for constant acceleration a Acceleration is slope of velocity vs time graph For constant velocity, a a 0 Instantaneous acceleration is a avg Δv dv dt lim Δ t 0 d dt 8
Recap Displacement is Δ Average velocity is v Δ avg, Instantaneous velocity is Average acceleration is a avg, Instantaneous acceleration is 0 + for constant velocity so 0 + v avg, v d dt Δv dv a d dt dt v 0 + a for constant acceleration a 9
One can derive other useful equations v 0 + a 0 + v 0 + 1 a No displacement in this equation ( ) No final velocity in this equation v v 0 + a ( ) 0 No time in this equation 0 + v 0 + No acceleration in this equation These formulas only work if the acceleration is constant! 10
Clicker question 3 Set frequency to BA The driver of a car going 30 m/s on a straight road spies a deer in the road 75 m ahead. We want to know what acceleration is needed to stop before hitting the deer? What formula, by itself, will give us the answer? A. B. C. D. v 0 + a + v 0 + 1 a v v 0 + a 0 0 + 1 ( ) ( ) ( v 0 + v ) E. None of the above 11
Solution: Using Eample problem 1 The driver of a car going 30 m/s on a straight road spies a deer in the road 75 m ahead. What acceleration is needed to stop before hitting the deer? First, draw a picture 30 m/s 75 m What do we know? v 0 30 m/s 0 a v v 0 ( 0 ) v v 0 + a f i ( ) ( ) 0 30 m/s 75 m 900 m /s 150 m 0 75 m we get acceleration: 6 m/s 1
Clicker question 3 Set frequency to BA The driver of a car going 30 m/s on a straight road spies a deer in the road 75 m ahead. We want to know what acceleration is needed to stop before hitting the deer? What formula, by itself, will give us the answer? A. B. C. D. v 0 + a + v 0 + 1 a v v 0 + a 0 0 + 1 E. None of the above ( ) ( ) ( v 0 + v ) These two equations can also be used to solve the problem. There is always more than one way to solve a problem. 13
Solution: Eample problem 1 (Solution ) What do we know? v 0 30 m/s 0 0 75 m Then solve 0 + 1 v 0 + ( ) gives v 0 + a 30 m/s to get a v 0 75 m ( 0 ) 1 v + v 0 ( ) 75 m 30 m/s 5 s 15 m/s 5 s 6 m/s Acceleration is negative because we defined velocity as positive and car is slowing 14
Clicker question 4 Set frequency to BA Usain Bolt accelerates at 4 m/s to his top speed of 1 m/s which he maintains until the finish. We want to find out how long he takes to run the 100 m dash. Which formula, by itself, will give us the answer? A. B. C. v 0 + a 0 + v 0 + 1 a 0 + 1 ( v + v ) 0 ( ) D. More than one of the above E. None of the above 15