An object's velocity is v⃗=c t^3 +d , where t is time and c and d are positive constants with ap

Name: Problem 21, Chapter 3: Essential University Physics
Uploaded: 2023-08-31T13:07:24-08:00
Duration: 2 min 35 s
Channel: Prabhat Tyagi
Description: An object's velocity is v⃗=c t^3 +d , where t is time and c and d are positive constants with appropriate units. What's the direction of the object's acceleration?

step 1 Here, in this question, we have given, velocity of object equals to C multiplied by T to the pow

Question

An object's velocity is $\vec{v}=c t^{3} \hat{\imath}+d \hat{\jmath},$ where $t$ is time and $c$ and $d$ are positive constants with appropriate units. What's the direction of the object's acceleration?

   An object's velocity is $\vec{v}=c t^{3} \hat{\imath}+d \hat{\jmath},$ where $t$ is time and $c$ and $d$ are positive constants with appropriate units. What's the direction of the object's acceleration?

Essential University Physics

Richard Wolfson 4th Edition

Chapter 3, Problem 21

Instant Answer

Solved by Expert Prabhat Tyagi

08/31/2023

Step 1

Acceleration is the derivative of velocity with respect to time. So, we differentiate the given velocity vector with respect to time: $$ \vec{a} = \frac{d\vec{v}}{dt} = \frac{d(ct^3 \hat{\imath} + d \hat{\jmath})}{dt} $$ Now, we differentiate each component Show more…

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An object's velocity is $\vec{v}=c t^{3} \hat{\imath}+d \hat{\jmath},$ where $t$ is time and $c$ and $d$ are positive constants with appropriate units. What's the direction of the object's acceleration?

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Key Concepts

Vector Kinematics

This concept involves the study of objects moving in space by analyzing vector quantities like velocity and acceleration. It is essential for understanding how an object’s position, speed, and direction vary over time.

Differentiation of Vector Functions

Differentiation is used to compute the acceleration vector from the velocity vector by taking the time derivative. This process helps determine how the components of velocity change with respect to time, which is fundamental for analyzing motion.

Unit Vectors

Unit vectors, typically denoted by symbols such as i, j, and k, are used to indicate direction in a coordinate system. They help decompose vector quantities into their directional components, making it easier to analyze the motion in each spatial dimension.

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