Suppose a marble rolls down a track after t seconds. Find the speed of the marble when t=2
The correct answer and explanation is:
To determine the speed of the marble when t=2t = 2, we need to consider how the position of the marble changes with respect to time. Typically, the motion of an object rolling down a track is governed by gravity and possibly friction. The general formula for the velocity (speed) of an object in free fall or down an incline under constant acceleration is given by: v(t)=atv(t) = at
where v(t)v(t) is the velocity at time tt, and aa is the constant acceleration.
For an object rolling down a track under the influence of gravity, the acceleration is typically a=gsin(θ)a = g \sin(\theta), where gg is the acceleration due to gravity (approximately 9.8 m/s29.8 \, m/s^2) and θ\theta is the angle of the track relative to the horizontal. However, if no angle is provided, assuming a frictionless track, the marble would be accelerating due to gravity alone, so the velocity formula would simplify to: v(t)=g⋅tv(t) = g \cdot t
Substituting g=9.8 m/s2g = 9.8 \, m/s^2 and t=2 st = 2 \, s, we get: v(2)=9.8×2=19.6 m/sv(2) = 9.8 \times 2 = 19.6 \, m/s
Thus, the speed of the marble after 2 seconds is 19.6 m/s19.6 \, m/s.
In this case, the assumption is that the marble is rolling without significant friction, and there is no other force acting on it except gravity. If the problem involves friction or a non-vertical incline, you would need to include additional factors like the coefficient of friction or the angle of the track in the acceleration calculation.