Physics Questions on Particle Motion

A particle moves along the x-axis so that at any time t ≥ 0 its position is given by $ x(t) = \frac{1}{2} (a - t)^2 $, where a is a positive constant. For what values of t is the particle moving to the right?

• The particle is moving to the right only if $ 0 < t < a $.

• The particle is moving to the right only if $ a < t $.

• The particle is moving to the right only if $ t = a $.

• The particle is not moving to the right.

A particle moves along the x-axis so that at time t ≥ 0 its position is given by $ x(t) = 2t^3 - 21t^2 + 72t - 53 $. At what time t is the particle at rest?

• $ t = 1 $ only

• $ t = 3 $ only

• $ t = 7/2 $ only

• $ t = 3 $ and $ t = 7/2 $

• $ t = 3 $ and $ t = 4 $

If the position of a particle on the x-axis at time t is $ -5t^2 $, then the average velocity of the particle for $ 0 \leq t \leq 3 $ is

• -45

• -30

• -15

• -10

• -5

For $ t \geq 0 $, the velocity of a particle moving along the x-axis is given by $ v(t) = t^3 - 6t^2 + 10t - 4 $. At what time t does the direction of motion of the particle change from right to left?

• 0.586

• 1.184

• 2.000

• 2.816

Two particles start at the origin and move along the x-axis. For $ 0 \leq t \leq 10 $, their respective position functions are given by $ x_1 = \sin t $ and $ x_2 = e^{-2t} - 1 $. For how many values of t do the particles have the same velocity?

• None

• One

• Two

• Three

• Four

• -4.071

• -2.048

• 0

• 5.153

• 6

• 0.387

• 0.831

• 1.243

• 1.647

• 8.094

A particle moves along a straight line with velocity given by $ v(t) = 5 + e^{t/3} $ for time $ t \geq 0 $. What is the acceleration of the particle at time $ t = 4 $?

• 0.422

• 0.698

• 1.265

• 8.794

• 28.381

A particle moves along the x-axis so that its position at time $ t $ is given by $ x(t) = t^2 - 6t + 5 $. For what value of $ t $ is the velocity of the particle zero?

• 1

• 2

• 3

• 4

• 5

The position of a particle moving along a straight line at any time $ t $ is given by $ s(t) = t^2 + 4t + 4 $. What is the acceleration of the particle when $ t = 4 $?