Graph Analysis and Motion Kinematics

Which x-t graph goes with the v-t graph?(Graph detail: When t = 0 s, x = –10 m.) A B C D OSU PH 211, Before Class #4

Which x-t graph goes with the v-t graph? Graph B(At every moment, the value of the velocity graph is the slope of the position graph.) A B C D OSU PH 211, Before Class #4

An object is accelerating toward a point. Evaluate the following statement (T/F/N):The object is getting closer and closer to that point. • Always True • Always False • Not enough information OSU PH 211, Before Class #4

An object is accelerating toward a point. Evaluate the following statement (T/F/N):The object is getting closer and closer to that point. • Always True • Always False • Not enough information An object’s acceleration is not necessarily in the same direction as its velocity. For example, the object could be moving away from the point but slowing down. OSU PH 211, Before Class #4

An object has a constant negative acceleration. 1. Its velocity must be positive. 2. Its velocity must be negative. 3. Its speed must be decreasing. 4. Its velocity is becoming more negative (or “less positive”). 5. Its velocity can never become zero. (What common, everyday event would demonstrate which of these is the true statement?) OSU PH 211, Before Class #4

An object has a constant negative acceleration. 1. Its velocity must be positive. 2. Its velocity must be negative. 3. Its speed must be decreasing. 4. Its velocity is becoming more negative (or “less positive”). 5. Its velocity can never become zero. (If we label downward as the negative vertical direction, then gravity’s downward acceleration is a good example of this.) OSU PH 211, Before Class #4

The Original Physics Lab The most universal of all experiences on Earth is the planet itself: its gravity. In free-fall, gravity treats every object the same. Near the surface of the Earth, for example, the acceleration in free-fall for any object is about 9.80 m/s2 (downward). Examples: Toss a ball upward at an initial velocity of (+)19.6 m/s. How fast is it moving 1 second later? 2 seconds? 3 seconds? Was the ball’s acceleration ever zero? (See After Class 4 for solutions.) A “slower” use of gravity helps reveal this (as it did for Galileo).…. OSU PH 211, Before Class #4

Describe the position, velocity, and acceleration time graphs for the following experiment: Motion Detector The Cart is given a brief “kick”. It travels up the ramp, momentarily comes to rest at the top of the ramp, and then rolls back down. We are graphing the motion from just after it left your hand until just before it was caught. OSU PH 211, Before Class #4

Position time Velocity time Acceleration time OSU PH 211, Before Class #4

Was the cart’s acceleration ever zero? No. The slope of the velocity graph doesn’t change as the velocity value passes through zero. OSU PH 211, Before Class #4

Standing in the middle of the MU quad, you throw a ball straight up in the air. At the ball’s highest point,… • vball = 0 and aball= 0 • vball ≠ 0 and aball= 0 • vball = 0 and aball≠ 0 • vball ≠ 0 and aball≠ 0 • Not enough information. OSU PH 211, Before Class #4

Standing in the middle of the MU quad, you throw a ball straight up in the air. At the ball’s highest point,… • vball = 0 and aball= 0 • vball ≠ 0 and aball= 0 • vball = 0 and aball≠ 0 • vball ≠ 0 and aball≠ 0 • Not enough information. Again, look at the graphs on slide 7. OSU PH 211, Before Class #4

Moving from graphs to equations By definition: a = dv/dt where v = dx/dt Now suppose: a is a constant over some time interval Dt We can get four 1-D (constant-acceleration) kinematics equations from these facts. (The book lists three equations, but there is a challenge in item 3 of Before Class 4: Can you derive the other? The solution will be in After Class 4.) And what about when acceleration is not constant? So long as we know how a varies with time, a(t), we can derive a useful set of equations. But the equations with constant acceleration are especially significant—because we live with it (gravity). OSU PH 211, Before Class #4

The Translational Kinematic Equations of Constant-Acceleration Motion vf = vi + at (doesn’t use x) x = vit + (1/2)a(t)2(doesn’t use vf) vf2 = vi2 + 2ax(doesn’t use t) x = (1/2)(vi + vf)t (doesn’t use a) Five variables (x, vi, vf, a and t): If you know any three, you can usually solve for the other two.Note: All variables except t are vector quantities; their signs (±) indicate their directions. OSU PH 211, Before Class #4

So: You look at your watch (that’s time ti) when the object is at position xi along your tape measure; you look at the watch again (at time tf) when the object is at position xf. (Then in the equations, use x = xf – xi and t = tf – ti) Again, caution: These equations apply to the motion of a rigid body only when a is constant. The object may have only one value for its acceleration vector throughout the entire time interval t. Whenever the value of a changes, you must start a fresh analysis.… OSU PH 211, Before Class #4

The kinematics equations of constant acceleration offer a tool like a picture frame that you set over all or part of a situation to compute an object’s motion. Example: You’re driving onto the freeway, accelerating ahead, and you suddenly see traffic stopped ahead. You move your foot to the brake and apply it steadily. What three different applications of the constant-acceleration kinematics equations must you make here—and why? OSU PH 211, Before Class #4

The kinematics equations of constant acceleration offer a tool like a picture frame that you set over all or part of a situation to compute an object’s motion. Example: You’re driving onto the freeway, accelerating ahead, and you suddenly see traffic stopped ahead. You move your foot to the brake and apply it steadily. What three different applications of the constant-acceleration kinematics equations must you make here—and why? Whenever the acceleration value changes, you must start a new calculation—using the final values from the previous calculation as the initial values for the new calculations. Here, there are three different values for the acceleration: speeding up, coasting (while moving your foot from gas to brake), braking. OSU PH 211, Before Class #4

Ball A is dropped from rest from a tall building.2 seconds later, ball B is dropped from rest from the same point. Once both balls are in motion: • Does their velocity difference increase, decrease, or remain the same with time? • Do they get further apart, closer together, or stay the same distance apart? Ignore air drag and assume a constant value of g (local free-fall acceleration). Could you answer these (and explain your reasoning) with either time graphs or kinematics equations? OSU PH 211, Before Class #4

Ball A is dropped from rest from a tall building. 2 seconds later, ball B is dropped from rest from the same point. When both balls are in motion: • Does their velocity difference increase, decrease, or remain the same with time? It remains the same, because earth’s constant gravitational acceleration is changing the velocity of each by the same constant amount (9.80 m/s downward) every second. Their v-t graphs are parallel lines, so the space between them never changes. • Do they get further apart, closer together, or neither? They get further apart, because the distance fallen is proportional to t2: (42 – 22) > (32 – 12) … (52 – 32) > (42 – 22) … (etc.) Their y-t graphs are parabolas offset horizontally by 2s, so the space between them increases with time. OSU PH 211, Before Class #4

A rocket lifts off (from rest) from the earth. During its boost phase, it has a vertically upward constant acceleration value aboost. At a time tb after lift-off, a bolt falls from the side of the rocket. (Probably not good.) Assuming no wind or air drag and a constant free-fall g value, draw the time graphs (a-t, v-t and y-t) for the bolt, from lift-off to impact. (See After Class 4 for the solution.) OSU PH 211, Before Class #4

Graph Analysis and Motion Kinematics