Equations
What equation gives velocity as a function of time under constant acceleration?
V(t) = V(o) + at
What does the slope of a velocity vs. time graph represent?
The acceleration
The vector quantity that specifies an object’s change in position.
Displacement
A velocity–time graph is a perfect horizontal line at v = +6 m/s. Identify the motion.
Uniform linear motion at 6 m/s to the right
You throw a ball straight up. At its highest point, velocity is zero—but is its acceleration zero?
No—acceleration equals g downward
How would you derive the position equation from the acceleration eq. using calculus and given initial condition for velocity?
1. Integrate a(t) with respect to time
2. Plug in initial condition for velocity (c value)
3. Integrate v(t) to find x(t)
A concave up position vs time graph with an increasing slope means what about the velocity and acceleration.
Both positive
The scalar rate of change of distance.
Speed
An acceleration–time graph is a horizontal line at a = –3 m/s². Identify the motion
Uniformly decelerated motion
A car goes around a circular track at constant speed. Is its velocity constant?
No: direction changes, so velocity vector changes
A car slows from 25 m/s to 5 m/s over 100 m. Derive the time it takes to do so.
1. V(f)^2 equation to find a
2. V(f) equation for Time
Given constant positive acceleration, describe the v(t) and x(t) graphs
V(t) is linearly increasing
X(t) is exponentially increasing (concave up parabola)
Motion under the influence of gravity only. Air resistance is not included.
Free-fall motion
A position–time graph is a downward‐opening parabola. Identify the motion (With regard to the acceleration)
Uniformly accelerated motion with constant negative acceleration
A train accelerates, then coasts at constant speed, then decelerates. Sketch qualitatively its position vs. time graph.
Curve upward (increasing slope), then straight line, then curve flattening out
A projectile is launched from a 45 m cliff at 20 m/s. Write the equation to find time it hits the ground.
~ 5.69 sec
A v(t) graph forms a triangle from 0 to 6 seconds, peaking at 10 m/s. What is its displacement.
30 meters
The component of acceleration tangent to the path in circular motion, responsible for changing the object’s speed along its trajectory.
Tangential acceleration
A velocity–time graph is a straight line with constant negative slope, crossing zero at t₀ and continuing into negative v. Identify the motion.
Motion reversing direction under constant acceleration (passing through v=0)
You drop one object and throw another horizontally from the same height at the same time. Which hits the ground first and which has the greater speed when it hits the ground?
Neither, they hit at the same time. The thrown object lands with a greater speed.
A particle has acceleration a(t) = 6t-12 and initial velocity v(o) = 10 Find the total distance traveled from t=0 to t=6
64.2 OR 64.3 meters
— find velocity with initial condition
— find zeros of v(t)
— 3 definite integrals considering positive/negative velocity
If s(t) is twice differentiable and at t₀ you have s″(t₀) = 0 and s‴(t₀) > 0, what features appear on the s–t, v–t, and a–t graphs at t₀?
An inflection point on the s-t graph, a local extremum on the v-t graph, and a zero crossing with positive slope on the a-t graph
The rate of change of acceleration, important for “smoothness” in motion.
jerk
A particle’s acceleration is always directed toward a fixed point and is proportional to its displacement from that point. Identify the motion,
Simple harmonic motion
Two projectiles are launched at the same speed but complementary angles θ and 90°–θ. Compare their ranges and maximum heights.
They have equal range; the one with the larger angle has the greater maximum height