Question

An object is moving along the x-axis, and its displacement ( x ) is given by the expression:

[ x = 30 + 20t - 15t² ]

where ( x ) is in meters (m) and ( t ) is in seconds (s).

(i) Find the expressions for the velocity (( v )) and acceleration (( a )).

(ii) Find the values of the initial position and initial velocity of the object.

Answer 1

Step-by-step explanation:

(i) Expressions for Velocity (v) and Acceleration (a):

Given the displacement function \(x = 30 + 20t - 15t^2\), we can find the velocity and acceleration by taking the first and second derivatives of the displacement function with respect to time (\(t\)).

Displacement (\(x\)):

\[x = 30 + 20t - 15t^2\]

Velocity (\(v\)):

\[v = \frac{dx}{dt} = 20 - 30t\]

Acceleration (\(a\)):

\[a = \frac{dv}{dt} = -30\]

(ii) Values of Initial Position and Initial Velocity:

To find the initial position (\(x_0\)) and initial velocity (\(v_0\)), we need to evaluate the displacement and velocity functions at \(t = 0\).

Initial Position (\(x_0\)):

\[x_0 = x(t=0) = 30\]

Therefore, the initial position (\(x_0\)) of the object is \(30\) meters.

Initial Velocity (\(v_0\)):

\[v_0 = v(t=0) = 20 - 30(0) = 20\]

Therefore, the initial velocity (\(v_0\)) of the object is \(20\) m/s.

Answer 2

The velocity and acceleration are given by v(t) = 20 - 30t m/s and a = -30 m/s². The initial position is 30 meters and the initial velocity is 20 meters per second.

Step-by-step explanation:

We're asked to find the expressions for velocity (v) and acceleration (a) for the given displacement equation, x(t) = 30 + 20t - 15t², and also to determine the values of the initial position and initial velocity of the object.

Step-by-Step Solution:

1. The velocity v is the first derivative of the displacement x with respect to time t, so v = dx/dt.
2. Differentiating x(t) = 30 + 20t - 15t² gives us v(t) = 20 - 30t.
3. Acceleration a is the derivative of velocity, so a = dv/dt.
4. Differentiating v(t) = 20 - 30t provides a(t) = -30; this confirms that acceleration is constant.
5. To find the initial position x0, we plug in t = 0 into the displacement equation, yielding x0 = 30 meters.
6. To find the initial velocity v0, we do the same for v(t), resulting in v0 = 20 meters per second.

Summary:

The expressions for velocity and acceleration are v(t) = 20 - 30t m/s and a = -30 m/s², respectively. The initial position is 30 meters, and the initial velocity is 20 meters per second.