Derivative of the Displacement Equation

Oct 13, 2025 by recoveryshake.com 142 views

The Displacement of a Particle Travels in a Straight Line in Second is Given by S=45t+11t²-t³: Find the Time When the Particle Comes to Rest

In physics, the displacement of a particle is a measure of the distance it travels from its initial position to its final position. The displacement of a particle can be described by a mathematical equation, which can be used to determine various properties of the particle's motion. In this article, we will discuss the displacement of a particle that travels in a straight line and is given by the equation S=45t+11t²-t³, where S is the displacement and t is the time. We will use this equation to find the time when the particle comes to rest.

The equation S=45t+11t²-t³ is a polynomial equation of degree 3, which means that it has three terms. The first term, 45t, represents the initial velocity of the particle, which is 45 units per second. The second term, 11t², represents the acceleration of the particle, which is 11 units per second squared. The third term, -t³, represents the deceleration of the particle, which is -1 unit per second cubed.

To find the time when the particle comes to rest, we need to find the time when the velocity of the particle is zero. The velocity of the particle is given by the derivative of the displacement equation with respect to time. Therefore, we need to find the derivative of the equation S=45t+11t²-t³.

Derivative of the Displacement Equation

To find the derivative of the displacement equation, we will use the power rule of differentiation, which states that if f(x) = x^n, then f'(x) = nx^(n-1). Applying this rule to the displacement equation, we get:

dS/dt = d(45t+11t²-t³)/dt = 45 + 22t - 3t²

To find the time when the particle comes to rest, we need to set the velocity to zero and solve for time. Therefore, we set the derivative of the displacement equation equal to zero and solve for t:

45 + 22t - 3t² = 0

The equation 45 + 22t - 3t² = 0 is a quadratic equation, which can be solved using the quadratic formula. The quadratic formula is given by:

t = (-b ± √(b² - 4ac)) / 2a

In this case, a = -3, b = 22, and c = 45. Plugging these values into the quadratic formula, we get:

t = (-(22) ± √((22)² - 4(-3)(45))) / 2(-3) = (-22 ± √(484 + 540)) / (-6) = (-22 ± √1024) / (-6) = (-22 ± 32) / (-6)

There are two possible solutions to the quadratic equation:

t = (-22 + 32) / (-6) = 10 / (-6) = -5/3

t = (-22 - 32) / (-6) = -54 / (-6) = 9

In this article, we discussed the displacement of a particle that travels in a straight line and is given by the equation S=45t+11t²-t³. We used this equation to find the time when the particle comes to rest by setting the velocity to zero and solving for time. We found two possible solutions to the quadratic equation, which are t = -5/3 and t = 9. However, since time cannot be negative, the only valid solution is t = 9.

The final answer is t = 9.
The Displacement of a Particle Travels in a Straight Line in Second is Given by S=45t+11t²-t³: Find the Time When the Particle Comes to Rest

Q&A: Understanding the Displacement of a Particle

A: The displacement of a particle is a measure of the distance it travels from its initial position to its final position.

A: The equation for the displacement of a particle in this problem is S=45t+11t²-t³, where S is the displacement and t is the time.

A: The initial velocity of the particle is 45 units per second, which is represented by the first term in the equation, 45t.

A: The acceleration of the particle is 11 units per second squared, which is represented by the second term in the equation, 11t².

A: The deceleration of the particle is -1 unit per second cubed, which is represented by the third term in the equation, -t³.

A: To find the time when the particle comes to rest, you need to find the time when the velocity of the particle is zero. This can be done by setting the derivative of the displacement equation equal to zero and solving for time.

A: The derivative of the displacement equation is dS/dt = 45 + 22t - 3t².

A: The quadratic equation can be solved using the quadratic formula, which is given by t = (-b ± √(b² - 4ac)) / 2a.

A: The two possible solutions to the quadratic equation are t = -5/3 and t = 9.

A: Since time cannot be negative, the only valid solution is t = 9.

A: The final answer is t = 9.

Not setting the derivative of the displacement equation equal to zero to find the time when the particle comes to rest.
Not using the quadratic formula to solve the quadratic equation.
Not considering the validity of the solutions to the quadratic equation.

In this article, we discussed the displacement of a particle that travels in a straight line and is given by the equation S=45t+11t²-t³. We used this equation to find the time when the particle comes to rest by setting the velocity to zero and solving for time. We also provided a Q&A section to help readers understand the concept better.