TPR Chapter 3: Kinematics

when the absolute value of the velocity is increasing, the object's speed

the object's speed is also increasing.

instantaneous velocity

the speed and direction of an object at a particular instant

A skydiver jumps out of a hovering helicopter. A few seconds later, another skydiver jumps out so they both fall with the same acceleration. Does the vertical distance between them (a) increase, (b) decrease or (c) stay the same? Does the difference in their velocities (d) increase, (e) decrease, or (f) stay the same? (Assume the g is constant.) Explain

(a) vertical distance increases. Because the first diver left earlier, He will always travel faster during his second than the second diver. Since he's traveling faster per second he travels further too. (f) stays the same because the acceleration, change in velocity is constant.

A rule of thumb for driving is that a separation of one car length for each 10 mph of speed should be maintained between moving vehicles. Assuming a constant reaction time, discuss the relevance of this rule for (a) motion with constant velocity and (b) motion with constant acceleration?

(a)For objects moving with constant velocity this rule of thumb is true. (b) however objects that move with constant acceleration experience continuous change in their velocity which means that the distance between cars will be changing as well.

A tennis player on serve tosses a ball straight up. As the tennis ball travels through the air, its speed (a) increases (b) decreases (c) increases and then decreases, (d) decreases and then increases, or (e) remain constant? Explain

(d) it decrease and then increases

A tennis player on serve tosses a ball straight up. While the ball is in free fall, does it's acceleration (a) increase (b) decrease (c) increase and then decrease, (d) decrease and then increase, or (e) remain constant? Explain

(e) remain constant... unless the ball is equipped with a motor or wings it has no ability to change it's acceleration. The acceleration the object will experience is that of gravitational acceleration.

subtracting vectors

1. reverse direction of the vector you want to subtract. 2. add as usual (it's the same as adding but with reversed).

equation for average velocity

v = ∆x/∆t = d/∆t. velocity = displacement / time.

Special Angles: √ 2, √ 3,

√ 2 = 1.4. (valentine s' day). √ 3 = 1.7. (st. Patrick's day).

distance

The length of a path between two points

Measurements: quantity, units, dimensions: speed, density, work,

speed (v): units = m/s; dimension = L/T. density (ρ): unit = kg/m^3; dimension = M/L^3. work (w) : unit = (kg*m^2)/S^2; dimensions = (M*L^2)/T^2

instantaneous acceleration

the change in an object's velocity at a specific instant of time

Displacement

the change in position + direction of an object. Vector quantity.

The direction of acceleration is the same as

the direction of the net force: ∆v if final velocity > initial velocity, then velocity increases and acceleration is positive. if if final velocity < initial velocity, then velocity decreases and acceleration is negative.

An object is speeding up when...

the initial velocity and acceleration have the same sign.

If acceleration is at an angle between 90 degrees and 180 degrees, then the speed is...

the speed is decreasing and the direction of velocity is changing. If the object's speed is decreasing then some of the acceleration must be pointing in the opposite direction of motion. So the angle between velocity and acceleration must be greater than 90 meaning 90 < θ < 180. The cosine of these angles is negative meaning the acceleration will point partially in the opposite direction of motion. (Think of an obtuse triangle).

Vector: definition and examples

A quantity that has both magnitude and direction. Examples of vector: displacement (m), velocity (m/s + direction), acceleration (m/s^2 + direction) force, momentum (N*m), friction.

You drop a ball from a window on an upper floor of a building. The ball strikes the ground with speed V. You now repeat the drop, but you have a friend down on the street who throws another ball upward at speed v. Your friend throws the ball upward at exactly the same time that you drop yours from the window. At some location the balls pass each other. Is this location at the halfway point between window and ground, above that point, or below that point?

Above the halfway point. The thrown pebble has an initial velocity and travels further than the dropped pebble which has no initial velocity. As a result the thrown pebble travel further.

On a position vs time graph, what does concave up and down mean?

Concave up means positive acceleration. Concave down means negative acceleration.

Distance vs. Displacement

Distance is a scalar quantity that refers to "how much ground an object has covered" during its motion. Displacement is a vector quantity that refers to "how far out of place an object is"; it is the object's overall change in position.

Acceleration

How fast an objects VELOCITY changes. Velocity = displacement / time. Therefore acceleration only exists if speed or direction changes.

When a ball is thrown upward... On its way down...

It loses acceleration (10m/s^2) every second. On its way down, it gains acceleration every second.

a cannon ball is shot from the ground level with an initial velocity of 100 m/s at an angle of 30 degrees from the ground. a). how high will the cannonball go? b). what is the ball's velocity at the top of the path? c). what will be the total flight time? d). how far will the cannonball travel horizontally?

Look art notebook for answers.

If a baseball has zero velocity at some instant, is the acceleration of the baseball necessarily zero?

No the acceleration does not have to be zero. Since acceleration is ∆v/t while velocity is only ∆x/∆t. For example, when you throw the baseball up in the air it has a constant acceleration of -9.8m/s^2 until it hits the ground. Now there is a point in the air where the baseball will quit going up and start to fall (velocity is zero) but acceleration is still -9.8m/s^2

If acceleration is zero, is the instantaneous acceleration also zero?

No, because the instantaneous velocity is changing direction.

Two cars are moving in the same direction in parallel lanes along a highway. At some instant, the velocity of car A exceeds the velocity of car B. Does this mean that the acceleration of A is greater than that of B? Explain

No. Car B may be picking up speed (that is, accelerating and have a higher acceleration) but have not reached the speed of car A yet. there are many other possible scenarios.

Is it possible for two cars to have the same velocity, but different speed?

No. Speed is the magnitude of the velocity vector. If velocities are the same, their magnitudes are the same, which is another way of saying that the speeds are the same.

A child throws a marble into the air with an initial speed Vo. Another child drops a ball at the same instant. Compare the accelerations of the two objects while they are in flight.

Once the objects leave the hand, they both have the same acceleration equal in magnitude to g gravity.

Trigonometric Functions

SOH CAH TOA.

Equation for average speed

Speed = distance/time: (s = d/t)

Velocity

The SPEED and DIRECTION of a moving object. "how fast can object's POSITION changes".

If a car is traveling eastward, can its acceleration be westward? Explain.

The car can be slowing down so yes.

If acceleration is perpendicular to velocity, then the object's speed is

The object's speed is constant, but it is still accelerating. This means that that the acceleration must point 90 degrees from the direction of motion (the velocity vector) -- cos 90 = 0 so none of acceleration points in the direction or opposite direction of motion. (Picture right triangle)

If acceleration is in the opposite direction of velocity, then, speed is

The object's speed is decreasing, but direction is not changing.

If acceleration is at an angle between 0 degree and 90 degrees, then the speed is...

The object's speed is increasing and the direction is changing. (Picture acute triangle).

A student at the top of a building of height h throws one ball upward with a speed of Vo and then throws a second ball downward with the same initial speed, Vo. How do the final velocities compare when the balls reach the ground?

The velocities will be the same. It will take the first ball longer to reach the ground but when it hits the ground it will have the same velocity.

Which among the following particles has an acceleration? a. A particle moving along a straight trajectory that is slowing down. b. A particle moving along a curved trajectory that has a constant speed. c.A particle moving along a curved trajectory that is speeding up.

They all have an acceleration, while the particle moving along a curved trajectory has constant speed, it still has an acceleration pointing to the center of the circle.

True or false: An object changing direction of motion experiences acceleration even when the object does not speed up or slow down.

True

The slope of (displacement) position vs time graph represents

Velocity. The steeper the slope, the higher the speed. If the slope is zero, velocity is zero and the object is at rest. If the slope is positive (linear line), velocity is positive and constant (zero acceleration). If the slope is negative (linear line), velocity is negative and constant (zero acceleration). if the slope is curved (not linear), velocity is NOT CONSTANT. The object is accelerating.

projectile motion equations: horizontal displacement

Vox = Vo cosine θ.

projectile motion equations: vertical displacement

Voy = Vo sine θ.

projectile motion equations: vertical velocity

Vy = Voy + (-g)t

If the speed of the particle is greater than zero, can the acceleration of the particle be zero?

Yes, acceleration is changing speeds, so as long as the speed is constant acceleration will be zero.

Can object have the same speed but different velocity?

Yes. Two objects can move at 10 m/s but travel in opposite directions (right and left for example) with respect to the point or axis of reference. Speed is a scalar but velocity is a vector. This is because the speed doesn't tell you in which direction the object is moving but the velocity tells you at what speed an object is moving and in which direction. So if your reference is the x-axis, the object moving at 10 m/s towards the right has a velocity of +10 m/s (or 10 m/s) and the one moving at 10 m/s towards the left has a velocity of -10 m/s.

Equation for average acceleration

a = ∆v/t. change in velocity/time. Velocity = displacement / time. Therefore acceleration only exists if speed or direction changes.

Unit Circle

a circle with a radius of 1, centered at the origin. Cosine: count down from 4 to 1 with increasing angle. 0: √4/2 = 1 30: √3/2 = 0.5 45:√2/2 = 0.7 60:√1/2 = 0.85 90: 0 180: - 1 Sine: count up from 1 to 4 with increasing angle. 0: 0 30: √1/2 = 0.5 45:√2/2 = 0.7 60:√3/2 = 0.85 90: √4/2 = 1 180: 0

scalar quantity

a quantity that can be described by magnitude only and has no direction. Examples: distance, time, speed, power, energy/work.

The slope of a velocity vs. time graph represents

acceleration. The steeper the slope, the greater the acceleration. If the slope is zero (flat line), the acceleration is zero. This means the object is at rest or has a constant velocity. If the slope is positive (linear line), acceleration is also positive and constant. If the slope is negative (linear line), then acceleration is also negative and constant. If the slope is curved, then acceleration is NOT CONSTANT.

zero acceleration

constant velocity (no change in speed or direction).

projectile motion equations: horizontal velocity

constant, because there is no acceleration in the horizontal direction.

The area under a velocity-time graph represents

displacement

Kinematic equation without initial velocity

displacement = (V*t) - 1/2(a*t^2)

Kinematic Equation without acceleration

displacement = 1/2 (Vi + Vf)t.

Equation for average distance

distance = speed x time. (d = m/s * s).

Equation for average displacement

final position - initial position. ∆x = Xf - Xi

Adding vectors

head to tail method. The tail of one vector must be at the head of the other vector.

Speed

how fast an object is moving

Measurements: SI units, measures, dimensions: meters, kilograms, seconds,

meters (m): measures length (L). kilograms (kg): measures mass (M). seconds (s): measures times (T).