To determine the velocity of a particle moving along a straight line with a given displacement equation, we can analyze the relationship between displacement and time. In this case, the displacement \( x \) is defined by the equation \( x = 3 + 5t \), where \( t \) represents time in seconds.
Understanding the Displacement Equation
The equation \( x = 3 + 5t \) can be broken down into two parts:
- Constant Term (3): This indicates the initial displacement of the particle when \( t = 0 \). So, at time zero, the particle is 3 meters away from the origin.
- Linear Term (5t): This part shows how the displacement changes with time. The coefficient of \( t \) (which is 5) represents the rate of change of displacement with respect to time.
Calculating Velocity
Velocity is defined as the rate of change of displacement with respect to time. Mathematically, this can be expressed as:
Velocity (v) = \(\frac{dx}{dt}\)
To find the velocity, we differentiate the displacement equation with respect to time:
Given:
\( x = 3 + 5t \)
Taking the derivative:
\( \frac{dx}{dt} = 0 + 5 = 5 \)
Interpreting the Result
The result indicates that the velocity of the particle is constant at 5 meters per second. This means that for every second that passes, the particle moves 5 meters along the straight line.
Summary
In summary, the velocity of the particle, derived from the displacement equation \( x = 3 + 5t \), is a uniform 5 meters per second. This consistent velocity suggests that the particle is moving steadily without any acceleration or deceleration.
