Physics 151 Week 5 Day 3

Topics– Motion with constant acceleration– Constant acceleration equations– Picture diagrams– Solving problems with constant acceleration

An arrow is launched vertically upward. It moves straight up to a maximum height, then falls to the ground. The trajectory of the arrow is noted. Which graph best represents the vertical velocity of the arrow as a function of time? Ignore air resistance; the only force acting is gravity.

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Answer

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Velocity to Position: Example 1

Find the position at times t = 1, 2, 3, 4, and 5 seconds.Assume x (t = 0 sec) = 0 m.

What is the displacement of the object between t = 1s and t = 3 s?

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Velocity to Position: Example 2

•Describe in words the motion of the object whose velocity graph is given below. What is happening at t = 2 s?

•Draw a motion diagram of the objects motion.

•Draw the corresponding position vs. time graph. Assume initial position is 5 m.

•Give an example of an object that would move this way

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Start your engine (velocity and acceleration)

On the on ramp to Interstate 25 a car accelerates at 7 m/s2 from rest.

What is the car's velocity at 1 second after it starts from rest?

2 seconds?

3 seconds?

4 seconds?Express your reasoning in words.

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Velocity Graph => Acceleration Graph

vx =slopeof xvs. t graph

ax = slope of vx vs. t graph

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x = area under vx vs. t graph

x=area=1

2b h =

1

2Δ t Δvx

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vx = area under ax vs. t graph

vx =area=bh=Δ t ax

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General Motion Model Graph Relationships

vx =slopeof xvs. t graph

ax = slope of vx vs. t graph

x=areaunder vx vs. t graph

Δvx = area under ax vs. t graph

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Constant Acceleration Equations

vx =area=bh=Δt axΔvx =ax Δt

⇒ v1x = v0 x + ax Δ t (equation 1)

x=areaunder vx vs. t

Δx = Area1 (square) + Area2 (triangle)

Δx = bh1 +1

2bh2 = Δ t v0 x +

1

2Δ t Δv1x

Δx = Δ t v0 x +1

2Δ t aΔt( )

⇒ Δx = v0 x

Δ t +1

2axΔ t 2 (equation 2)

Area 1

Area 2

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Constant Acceleration Equations (Equation 3)

⇒ v1x = v0 x + ax Δ t (equation 1)

⇒ x=v0 x Δ t +1

2ax Δ t 2 (equation 2)

ax t=v1x − v0 x (equation 1)

Δ t =v1 x − v0 x

ax

x=v0 x

v1 x − v0 x

ax

⎛

⎝⎜⎞

⎠⎟+

1

2ax

v1x − v0 x

ax

⎛

⎝⎜⎞

⎠⎟

2

(equation 2)

x=v0 xv1x

ax−v0 x

2

ax+

1

2

v1x2 − 2v0 xv1x + v0 x

2

ax

⎛

⎝⎜⎞

⎠⎟

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Constant Acceleration Equations (Equation 3)

x=v0 xv1x

ax−v0 x

2

ax+

1

2

v1x2 − 2v0 xv1x + v0 x

2

ax

⎛

⎝⎜⎞

⎠⎟

2axx=2v0 xv1x −2v0x2 + v1x

2 −2v0xv1x + v0x2( )

2axx=2v0 xv1x −2v0xv1x −2v0x2 + v0x

2 + v1x21x =v1x

21x −v0x

2

v1 x21x −v0x

2 =2axx

⇒ v1x21x =v0x

2 + 2axx equation3

Unit Check => ???

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MCAT Problem

A constantly accelerating particle starts from rest and travels 50 m. If it reaches a velocity of 100 m/s, how long did the 50 m trip take?

A. 0.5 s

B. 1.0 s

C. 2.0 s

D. 3.0 s

Pictorial Diagrams

Pictorial Diagram Example 1

Pictorial Diagram Example 2

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Hints for Motion Problems

1. Start with a diagram or graph and break the motion into motion intervals

2. Identify the motion models that pertain to each interval.

3. Think about what motion relationships exist in the problem situation.

4. For constant a motion, write out the 3 equations and underline what you know and circle what you are looking for. This can help suggest how to approach the problem.

5. Do your algebra in symbols. It will make problem solving easier.

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The Sprinter

A sprinter accelerates at 2.5 m/s^2 until reaching his top speed of 15 m/s. He then continues to run at top speed.

How long does it take him to run the 100 m dash?

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Down and Up

A ball is released on the left side at a height of 1 m on a frictionless 30 degree slope, |a| = 5 m/s2. At the bottom, it turns smoothly onto a 60 degree slope going back up, |a| = 8.66 m/s2.

What maximum height does it reach on the right side?

Timing the light

You have a summer job working for the UNM police department studying traffic flow on and near campus. There have been several complaints concerning the signal on Yale and Central Avenue for drivers on Central. The complaints claim that the yellow light is too short. If most cars decelerate at 10.0 m/s2 while braking, how long should the light be so that people who can’t stop before they go into the intersection after the signal turns yellow have time to go through the 16.0 m wide intersection?

Exam Tips

• Instantaneous velocity

• Area under the graph

• Algebra in symbols

• Givens Required

• Complete explanations

• Use homework to practice good solutions