The position of a particle as it moves along a y axis is given by y=(2.0 cm) sin(πt / 4) with t

step 1 In this problem of motion along a straight line we have given that. The position of a particle a

The position of a particle as it moves along a y axis is...

Key Concepts

Differentiation in Kinematics

Differentiation is a mathematical tool used in kinematics to determine rates of change, such as velocity and acceleration, from a position-time function. By applying calculus, specifically the chain rule when necessary, one can derive the instantaneous velocity as the first derivative and the instantaneous acceleration as the second derivative of the displacement function.

Average Acceleration

Average acceleration measures the change in velocity over a given time interval, calculated by subtracting the initial velocity from the final velocity and dividing by the elapsed time. It represents how quickly velocity changes overall during the interval and provides a broad view of the object's change in motion.

Instantaneous Acceleration

Instantaneous acceleration is the rate at which velocity is changing at a particular moment in time. It is obtained by differentiating the velocity function with respect to time (or by taking the second derivative of the position function with respect to time). This concept describes how fast the velocity of the object is increasing or decreasing at any specific point.

Average Velocity

Average velocity refers to the total displacement divided by the total time over an interval. It is a measure of an object's overall change in position, regardless of any variations in speed or direction during that period. In kinematics, it is computed as the difference between the final and initial positions, divided by the time elapsed.

Instantaneous Velocity

Instantaneous velocity is the speed and direction of an object at a specific moment in time. It is found by taking the derivative of the position function with respect to time. This concept captures the precise rate of change of position at any given instant, reflecting the object's motion at that specific moment.

Transcript

00:01 In this problem of motion along a straight line we have given that.

00:04 The position of a particle as it moves along a y -axis is given y, y is equal 2 .0 sign.

00:14 This is sine pi multiplied with t, so pi t divided with 4.

00:19 Where this is in centimeter so we can write it as more clear.

00:24 This is 2 .0 centimeter and multiplied to sine pi t divided with 4.

00:28 Now with t in second and y is centimeter so t is in second and y is in centimeters we have to find what is the average velocity of the particle between t is equal to t is 0 to 2 second so average velocity so v average so changing position so why final position would be 2 seconds minus y initial so this is 0 divided with time taken so this is delta t is equal to 2 second so velocity average would be equal to so sign this value would be pi multiply with 2 this is pi divided with 2 is 2 .0 0 minus sine 0 is equal to 0 so this is 0 delta t is equal to 2 seconds so this is equal to 1 .0 centimeters per second this is the every velocity or the solution for part a now part b in part b it is asked that what is the instantaneous velocity at t is equal to 0 1 and 2 seconds so for this we have to differentiate it so velocity is given as differentiation of sign is cost so this is 2 .0 and multiplied with this angle so this value would be pi divide with 4 and cos pi t divided with 4 when we more solve it so this is equal to pi divided with 2 cos pi t divide with 4 now we want the velocity at t is equal to 0 so we have to find v0 t is equal to 0 that means cost 0 is equal to 1 so we have v0 is equal to pi divide with 2 centimeters per second now t is equal to 1 so v1 putting the value t is equal to 1 so pi divide with 4 is equal to here this is root 2 divide with 2 so we have to multiply it so this value would be pi root 2 divide with 4 centimeters meters per second.

02:35 Now velocity at t is equal to 2 second.

02:39 So here this value would be v2.

02:44 Now putting the value t is equal to 2 pi divided with 4 which is equal to cost pi divide with 2 and cos pi divide with 2 is equal to 0.

02:52 So 5 divided with 2 multiplied with 0 is equal to 0.

02:56 So we have the right answer as 0 centimeters per second.

03:00 So this is the answer for part b.

03:04 Now part c.

03:07 Part c says, what is the average acceleration of the particle between t is equal to 0 to 2 seconds? so first we have to find the velocity, then acceleration.

03:17 So we have the average acceleration say average acceleration between t is equal to 2 second to say a2 or we can say here change in velocity so here we would have to put velocity at time 2 seconds.

03:35 So, here v2 minus v1 divide with the time taken.

03:39 So v2 is equal to 0 minus v1 is equal to pi divide with 2 divided with 2 seconds.

03:47 So this is equal to minus pi divide with 4 centimeters per second square.

03:52 So we have the answer which is equal to minus pi divide with 4 centimeters per second square.

04:00 This is the solution for part c...

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