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The displacement of a particle is given by s=3t2+2t. What is the velocity of the particle at t=3 seconds?
18 m/s
20 m/s
24 m/s
21 m/s
20 m/s
The velocity of a particle at any instant is defined as the instantaneous rate of change of displacement with respect to time. By differentiating the given displacement equation s=3t2+2t with respect to time t, we obtain the velocity function v=dtdsтАЛ.
The velocity of a particle at any instant is defined as the instantaneous rate of change of displacement with respect to time. By differentiating the given displacement equation s=3t2+2t with respect to time t, we obtain the velocity function v=dtdsтАЛ.
Think of the speedometer in your car; it shows your current speed at a specific moment rather than your average speed over the entire trip, which is exactly what the derivative calculates.
v=dtdsтАЛ тАФ Velocity as the derivative of displacement
v=6t+2 тАФ The velocity function for the given displacement
According to the principles of kinematics, velocity is the first derivative of displacement. By applying the power rule of differentiation, dtdтАЛ(atn)=antnтИТ1, we calculate v=dtdтАЛ(3t2+2t)=6t+2. Substituting t=3 gives v=6(3)+2=20┬аm/s.
Instantaneous velocity is calculated by evaluating the derivative at a specific time value.
The power rule is the fundamental tool for differentiating polynomial motion equations.
Dimensional consistency requires that velocity units be in m/s given s in meters and t in seconds.
Calculus provides precise values for non-uniform motion.
Requires knowledge of differentiation for non-linear displacement functions.
Calculating instantaneous speed in traffic accidents.
Analyzing rocket trajectory profiles.
The displacement s=3t2+2t represents uniformly accelerated motion because the second derivative, acceleration, is constant at a=6┬аm/s2.
Option A (18 m/s) results from forgetting the constant derivative term, and Option C (24 m/s) results from incorrectly evaluating the expression.
B is correct тАФ The velocity is determined by finding the derivative of the displacement function and evaluating it at t=3, yielding 6(3)+2=20┬аm/s.
If you see a time-dependent displacement function s(t), always differentiate to find velocity and differentiate again to find acceleration.
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