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3.6 Finding Velocity and Displacement from Acceleration

3 min read•june 24, 2024

Kinematics with is all about predicting using math. We use equations to figure out how things move when they speed up or slow down at a steady rate.

These equations help us calculate position, , and over . They're super useful for real-world problems, like figuring out how far a car will travel or how high a ball will go when thrown.

Kinematics with Constant Acceleration

Kinematic equations from calculus

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  • Acceleration represents the rate of change of velocity with respect to time and is expressed as a=dvdta = \frac{dv}{dt}
  • Velocity represents the rate of change of position with respect to time and is expressed as v=dxdtv = \frac{dx}{dt}
  • Velocity as a function of time is derived by integrating acceleration with respect to time resulting in v(t)=∫adt=at+v0v(t) = \int a dt = at + v_0, where v0v_0 represents the initial velocity (at t=0t=0)
  • Position as a function of time is derived by integrating velocity with respect to time resulting in x(t)=∫v(t)dt=∫(at+v0)dt=12at2+v0t+x0x(t) = \int v(t) dt = \int (at + v_0) dt = \frac{1}{2}at^2 + v_0t + x_0, where x0x_0 represents the initial position (at t=0t=0)

Application of kinematic equations

  • The equation v(t)=v0+atv(t) = v_0 + at is used to find velocity at a specific time by substituting known values for initial velocity v0v_0, acceleration aa, and time tt (car accelerating from rest)
  • The equation x(t)=x0+v0t+12at2x(t) = x_0 + v_0t + \frac{1}{2}at^2 is used to find position at a specific time by substituting known values for initial position x0x_0, initial velocity v0v_0, acceleration aa, and time tt (ball thrown upwards)
  • The equation vf2=v02+2a(xf−x0)v_f^2 = v_0^2 + 2a(x_f - x_0) relates final velocity vfv_f, initial velocity v0v_0, acceleration aa, and xf−x0x_f - x_0 and is derived by substituting tt from v(t)=v0+atv(t) = v_0 + at into x(t)=x0+v0t+12at2x(t) = x_0 + v_0t + \frac{1}{2}at^2 (braking distance of a car)
  • These equations describe the motion of objects and their over time

Velocity functions from acceleration

  • For constant acceleration, the equation v(t)=v0+atv(t) = v_0 + at is used by substituting the given constant acceleration value for aa and including the initial velocity v0v_0 if provided, otherwise assuming v0=0v_0 = 0 (object falling under gravity)
  • For acceleration as a function of time, the velocity function is obtained by integrating the acceleration function with respect to time using v(t)=∫a(t)dt+v0v(t) = \int a(t) dt + v_0, evaluating the integral and adding the initial velocity v0v_0 if provided, otherwise assuming v0=0v_0 = 0 (rocket with varying thrust)
  • , such as initial velocity and position, are crucial for determining the complete velocity function

Position functions from velocity

  • For constant velocity, the equation x(t)=x0+vtx(t) = x_0 + vt is used by substituting the given constant velocity value for vv and including the initial position x0x_0 if provided, otherwise assuming x0=0x_0 = 0 (train moving at constant speed)
  • For velocity as a function of time, the position function is obtained by integrating the velocity function with respect to time using x(t)=∫v(t)dt+x0x(t) = \int v(t) dt + x_0, evaluating the integral and adding the initial position x0x_0 if provided, otherwise assuming x0=0x_0 = 0 (object thrown with varying velocity)

Vector and Scalar Quantities in Kinematics

  • Velocity and acceleration are , having both magnitude and direction
  • Time and are , having only magnitude
  • Understanding the distinction between vector and scalar quantities is essential for correctly applying

Key Terms to Review (24)

Acceleration: Acceleration is the rate of change of velocity with respect to time. It represents the change in an object's speed or direction over a given time interval, and is a vector quantity that has both magnitude and direction.
Brownian motion: Brownian motion is the random, erratic movement of particles suspended in a fluid (liquid or gas) resulting from collisions with fast-moving molecules of the fluid. It provides evidence for the kinetic theory of gases and supports the concept of molecular motion.
Constant Acceleration: Constant acceleration refers to a state of motion where the rate of change in velocity remains the same over time. This concept is fundamental in understanding the behavior of objects under the influence of a constant force, as described in the topics of finding velocity and displacement from acceleration, as well as projectile motion.
Differentiation: Differentiation is the process of finding the derivative of a function, which represents the rate of change of the function at a specific point. It is a fundamental concept in calculus that is used to analyze the behavior of functions and solve a wide range of problems in physics, engineering, and other scientific fields.
Displacement: Displacement is a vector quantity that refers to the change in position of an object. It is measured as the straight-line distance from the initial to the final position, along with the direction.
Displacement: Displacement is the change in position of an object relative to a reference point. It is a vector quantity, meaning it has both magnitude and direction, and is used to describe the movement of an object in physics.
Elapsed time: Elapsed time is the total duration taken for an event to occur, measured from its start to its end. It is a scalar quantity typically measured in seconds, minutes, or hours.
Free fall: Free fall is the motion of an object under the influence of gravitational force only. It neglects air resistance and assumes a uniform acceleration due to gravity.
Free Fall: Free fall is a state of motion where an object is falling under the sole influence of gravity, without any other external forces acting upon it. This term is closely connected to the topics of motion with constant acceleration, projectile motion, Newton's second law, and gravitational effects near Earth's surface.
Initial Conditions: Initial conditions refer to the starting values or parameters that define the state of a system at the beginning of an analysis or problem-solving process. These starting values are crucial in determining the subsequent behavior and evolution of the system over time.
Integration: Integration is a fundamental mathematical operation that involves finding the area under a curve or the accumulation of a quantity over an interval. It is the inverse operation of differentiation and is essential in various fields, including physics, engineering, and economics.
Kinematic Equations: Kinematic equations are a set of mathematical relationships that describe the motion of an object, including its position, velocity, and acceleration, without considering the forces that cause the motion. These equations are fundamental in the study of classical mechanics and are widely used in the analysis of various types of motion, such as free fall, projectile motion, and uniform acceleration.
Kinematics Equations: Kinematics equations are a set of mathematical relationships that describe the motion of an object, particularly its position, velocity, acceleration, and time. These equations are fundamental to the study of motion in physics and are essential for understanding and analyzing various types of motion, including constant acceleration motion and the determination of velocity and displacement from acceleration.
Meters per Second Squared: Meters per second squared (m/s²) is a unit of acceleration, which measures the rate of change in velocity over time. It represents the change in velocity, in meters per second, that occurs in one second. This unit is fundamental in understanding the concepts of motion, force, and gravity in physics.
Motion: Motion refers to the change in position of an object over time. It is a fundamental concept in physics that describes the movement of objects in space and the factors that influence their behavior. Motion is a crucial component in understanding various physical phenomena, including the motion of celestial bodies, the motion of everyday objects, and the motion of subatomic particles.
Position-Time Graph: A position-time graph is a graphical representation that shows the position of an object as a function of time. It is a fundamental tool used to analyze and understand the motion of an object in physics, particularly in the context of kinematics, which is the study of motion without considering the forces that cause it.
Scalar Quantities: Scalar quantities are physical quantities that are fully described by a single numerical value and a unit, without the need for any directional information. They have magnitude but no direction.
Time: Time is a fundamental concept in physics that represents the duration or interval between events, the order in which they occur, and the measurement of their rate of change. It is a crucial factor in understanding the physical world and the laws that govern it.
Trajectory: A trajectory is the path that a projectile follows through space as a function of time. It is determined by initial velocity, launch angle, and the forces acting on the projectile, such as gravity and air resistance.
Trajectory: Trajectory refers to the path or curve that an object follows through space over time. It describes the motion and position of an object as it moves under the influence of various forces, such as gravity, air resistance, and initial velocity.
Uniform Acceleration: Uniform acceleration refers to a constant rate of change of velocity of an object over time, meaning that the acceleration does not vary. This concept is crucial in analyzing the motion of objects under consistent forces, allowing for predictable changes in velocity and displacement. With uniform acceleration, the equations of motion can be applied straightforwardly to determine various parameters like velocity and displacement.
Vector Quantities: Vector quantities are physical quantities that have both magnitude and direction, distinguishing them from scalar quantities, which only have magnitude. In physics, understanding vector quantities is crucial for analyzing motion and forces, as they provide essential information about how objects move and interact in space.
Velocity: Velocity is a vector quantity that describes the rate of change of an object's position with respect to time. It includes both the speed and the direction of an object's motion, making it a more complete description of an object's movement compared to just speed alone.
Velocity-Time Graph: A velocity-time graph is a graphical representation that depicts the relationship between an object's velocity and time. It is a fundamental tool in understanding and analyzing the motion of an object, as it provides a visual representation of the object's speed and direction of motion over a given time period.
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