# Particle Motion; Acceleration directly proportional to time

• Alexanddros81
In summary, the problem involves a particle with an acceleration that is directly proportional to time. At t = 0, the velocity of the particle is 16 in./s. Given that v = 15 in./s and x = 20 in. when t = 1 s, the velocity, position, and total distance traveled when t = 7s need to be determined. The solution involves using the equation v-v_0 = \int_0^t a \, dt + C1, where a is directly proportional to t, to find the value of C1 and then using that value to solve for the velocity, position, and total distance traveled at t = 7s.
Alexanddros81

## Homework Statement

11.10 The acceleration of a particle is directly proposional to the time t.
At t = 0, the velocity of the particle is v = 16 in./s. Knowing that v = 15 in./s
and that x = 20 in. when t = 1 s, determine the velocity, the position, and
the total distance traveled when t = 7s.

## The Attempt at a Solution

Totaly confused on this one; Any hints?

#### Attachments

• Vector Mechanics Dynamics Beer P11_10 s.jpg
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The acceleration depends on time in this problem. You need to use that information.

Hi!
Is the following correct?

##v-v_0 = \int_0^1 a \, dt + C1##
##v-v_0 = \int_0^1 f(t) \, dt + C1##
Since a is directly propotional to t it follows that
##v-v_0 = \int_0^1 t \, dt + C1##
##v-v_0 = \left[ \frac {t^2} {2} \right]_0^1 + C1##
##v-v_0 = \frac {1} {2} + C1##
##15-16 = \frac {1} {2} + C1 \Rightarrow C1 = -\frac {3} {2}##

Is ## a = -1 \frac {in} {s^2}## as I have calculated on my first post?

## What is particle motion?

Particle motion is the movement of a small, localized object through space. It can refer to the motion of atoms, molecules, or other small particles.

## What is acceleration directly proportional to time?

Acceleration directly proportional to time means that as time increases, the acceleration of an object also increases at a constant rate. This relationship can be represented mathematically as a linear function, where acceleration (a) is equal to the time (t) multiplied by a constant (k): a = kt.

## What is the difference between speed and acceleration?

Speed is the rate at which an object is moving, while acceleration is the rate at which an object's speed is changing. Speed is a scalar quantity, meaning it only has magnitude, while acceleration is a vector quantity, meaning it has both magnitude and direction.

## How is acceleration measured?

Acceleration is typically measured in meters per second squared (m/s^2) in the metric system, or feet per second squared (ft/s^2) in the imperial system. It can also be measured using other units such as kilometers per hour squared or miles per hour squared.

## What factors can affect particle motion and acceleration?

Some factors that can affect particle motion and acceleration include the force acting on the object, the mass of the object, and any external factors such as friction or air resistance. In addition, the initial velocity and direction of the object can also impact its motion and acceleration.

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