Kinematics Practice Test

1.

A particle moves along the x-axis so that its position is given by \( x(t) = 3t^3 - 2t^2 + t \) (in meters, t in seconds). What is the particle's acceleration at \( t = 2 \) s?

a) 32 m/s² b) 28 m/s² c) 22 m/s² d) 16 m/s²

2.

A projectile is launched from ground level with speed \( v_0 \) at angle \( \theta \) above the horizontal. Derive an expression for the total time of flight, neglecting air resistance.

a) \( \frac{2v_0 \sin\theta}{g} \) b) \( \frac{v_0 \sin\theta}{g} \) c) \( \frac{2v_0 \cos\theta}{g} \) d) \( \frac{v_0^2 \sin2\theta}{g} \)

3.

A car accelerates from rest with a time-dependent acceleration \( a(t) = 6t \) m/s². What is its velocity after 4 seconds?

a) 48 m/s b) 24 m/s c) 12 m/s d) 36 m/s

4.

A ball is thrown vertically upward with initial velocity \( v_0 \). At what height (above the ground) is its speed exactly half of what it will be just before hitting the ground?

a) \( \frac{3}{4}h_{max} \) b) \( \frac{1}{2}h_{max} \) c) \( \frac{1}{4}h_{max} \) d) \( \frac{2}{3}h_{max} \)

5.

A particle moves in the plane with velocity \( \vec{v}(t) = (4t)\hat{i} + (3)\hat{j} \) m/s. If it starts at the origin, what is its position vector at \( t = 2 \) s?

a) \( (8\hat{i} + 6\hat{j}) \) m b) \( (16\hat{i} + 6\hat{j}) \) m c) \( (4\hat{i} + 3\hat{j}) \) m d) \( (8\hat{i} + 3\hat{j}) \) m

6.

A train moves with constant acceleration. It passes point A with speed 10 m/s and point B, 200 m further, with speed 30 m/s. What is the acceleration?

a) 4 m/s² b) 2 m/s² c) 1 m/s² d) 0.5 m/s²

7.

A particle starts from rest and moves along a straight line with acceleration \( a(x) = 6x \) m/s². What is its speed when it has traveled 2 m?

a) 4.9 m/s b) 6.0 m/s c) 3.5 m/s d) 2.4 m/s

8.

A projectile is launched from height \( h \) above the ground with initial speed \( v_0 \) at angle \( \theta \). Derive an expression for the horizontal range.

a) \( \frac{v_0 \cos\theta}{g} \left(v_0 \sin\theta + \sqrt{(v_0 \sin\theta)^2 + 2gh}\right) \) b) \( \frac{v_0^2 \sin2\theta}{g} \) c) \( \frac{2v_0^2 \sin\theta \cos\theta}{g} \) d) \( \frac{v_0 \sin\theta}{g} \)

9.

A particle moves so that its velocity is given by \( v(t) = At^2 + B \), where \( A \) and \( B \) are constants. If \( x = 0 \) at \( t = 0 \), what is its position at time \( t \)?

a) \( \frac{A}{3}t^3 + Bt \) b) \( \frac{A}{2}t^2 + Bt \) c) \( At^3 + B \) d) \( At^2 + Bt \)

10.

A boat heads due east across a river with speed 5 m/s relative to the water. The river flows north at 3 m/s. What is the magnitude of the boat's velocity relative to the ground?

a) 5.83 m/s b) 8.00 m/s c) 2.00 m/s d) 3.00 m/s

11.

A particle moves along the y-axis so that its velocity is given by \( v(t) = 2e^{0.5t} \) m/s. If \( y = 0 \) at \( t = 0 \), what is its position at \( t = 2 \) s?

a) 6.87 m b) 5.16 m c) 4.00 m d) 2.72 m

12.

A ball is dropped from a height \( h \) above the ground. At what height is its speed exactly half of what it will be just before hitting the ground?

a) \( \frac{3}{4}h \) b) \( \frac{1}{2}h \) c) \( \frac{1}{4}h \) d) \( \frac{2}{3}h \)

13.

A particle moves so that its acceleration is \( a(t) = 12t^2 \) m/s². If its initial velocity is zero, what is its displacement after 2 seconds?

a) 16 m b) 24 m c) 32 m d) 48 m

14.

A projectile is launched at speed \( v_0 \) from a cliff of height \( h \) above level ground at angle \( \theta \) above the horizontal. Which of the following best describes the time to reach the ground?

a) The positive root of \( y(t) = h + v_0 \sin\theta\, t - \frac{1}{2}gt^2 = 0 \) b) \( \frac{2v_0 \sin\theta}{g} \) c) \( \frac{v_0 \sin\theta}{g} \) d) \( \frac{v_0^2 \sin2\theta}{g} \)

15.

A particle moves along a straight line with velocity \( v(x) = 2x \) m/s, where \( x \) is in meters. If it starts from \( x = 1 \) m at \( t = 0 \), how long does it take to reach \( x = 4 \) m?

a) 0.69 s b) 1.10 s c) 2.00 s d) 4.00 s

16.

A car moves such that its velocity is \( v(t) = 10 - 2t \) m/s. At what time does it come to rest?

a) 5 s b) 10 s c) 2 s d) 20 s

17.

A particle moves in a circle of radius 2 m such that its tangential acceleration is constant at 3 m/s². If it starts from rest, what is its speed after completing one full revolution?

a) 8.7 m/s b) 6.9 m/s c) 4.9 m/s d) 3.5 m/s

18.

A particle moves along the x-axis with acceleration \( a(x) = -kx \), where \( k > 0 \). What type of motion does the particle undergo?

a) Simple harmonic motion b) Uniform acceleration c) Exponential decay d) Projectile motion

19.

A projectile is launched horizontally from a height \( h \) with speed \( v_0 \). How long does it take to reach the ground?

a) \( \sqrt{\frac{2h}{g}} \) b) \( \frac{h}{v_0} \) c) \( \frac{v_0}{g} \) d) \( \sqrt{\frac{h}{g}} \)

20.

A car travels 1 km at 20 m/s and then another 1 km at 40 m/s. What is its average speed for the entire trip?

a) 26.7 m/s b) 30.0 m/s c) 32.0 m/s d) 28.0 m/s

21.

A particle moves along the x-axis with velocity \( v(t) = 5\cos(2t) \) m/s. If \( x = 0 \) at \( t = 0 \), what is its position at \( t = \frac{\pi}{4} \) s?

a) 2.5 m b) 1.8 m c) 3.5 m d) 0.0 m

22.

A ball is thrown downward from a height \( h \) with initial speed \( v_0 \). What is its speed just before hitting the ground?

a) \( \sqrt{v_0^2 + 2gh} \) b) \( v_0 + gh \) c) \( v_0 + 2gh \) d) \( \sqrt{2gh} \)

23.

A particle moves so that its acceleration is \( a(t) = -4\sin(2t) \) m/s². If its initial velocity is zero, what is its velocity at \( t = \frac{\pi}{2} \) s?

a) -4 m/s b) 0 m/s c) 2 m/s d) 4 m/s

24.

A projectile is launched at speed \( v_0 \) at angle \( \theta \) above the horizontal. What is the maximum height reached?

a) \( \frac{v_0^2 \sin^2\theta}{2g} \) b) \( \frac{v_0^2 \sin2\theta}{2g} \) c) \( \frac{v_0^2 \cos^2\theta}{2g} \) d) \( \frac{v_0^2}{2g} \)

25.

A particle moves along a straight line so that its velocity is \( v(x) = 3x^2 \) m/s. If it starts from rest at \( x = 0 \), what is its acceleration when \( x = 2 \) m?

a) 144 m/s² b) 36 m/s² c) 24 m/s² d) 18 m/s²

26.

A car accelerates from rest with constant acceleration \( a \) for time \( t_1 \), then moves at constant speed for time \( t_2 \), and finally decelerates at rate \( a \) to rest. If the total distance is \( d \), derive an expression for \( t_2 \).

a) \( t_2 = \frac{d}{a t_1} - t_1 \) b) \( t_2 = \frac{d}{2 a t_1} \) c) \( t_2 = \frac{d}{a t_1} + t_1 \) d) \( t_2 = \frac{d}{a t_1^2} \)

27.

A particle moves such that its position is \( x(t) = A \cos(\omega t) \). What is its maximum acceleration?

a) \( A \omega^2 \) b) \( A \omega \) c) \( \omega^2 \) d) \( A \)

28.

A projectile is launched at speed \( v_0 \) at angle \( \theta \) above the horizontal. What is the total horizontal distance (range) if it lands at the same height?

a) \( \frac{v_0^2 \sin2\theta}{g} \) b) \( \frac{2v_0^2 \sin\theta \cos\theta}{g} \) c) \( \frac{v_0^2 \cos^2\theta}{g} \) d) \( \frac{v_0^2}{g} \)

29.

A particle moves along the x-axis with velocity \( v(x) = kx \), where \( k > 0 \). If it starts from \( x_0 \) at \( t = 0 \), what is its position as a function of time?

a) \( x(t) = x_0 e^{kt} \) b) \( x(t) = x_0 + kt \) c) \( x(t) = x_0 e^{-kt} \) d) \( x(t) = x_0 + k t^2 \)

30.

A car moves with velocity \( v(t) = v_0 e^{-\alpha t} \), where \( v_0 \) and \( \alpha \) are constants. What is the total distance traveled as \( t \to \infty \)?

a) \( \frac{v_0}{\alpha} \) b) \( v_0 \alpha \) c) \( v_0 \) d) \( \frac{1}{v_0 \alpha} \)

Kinematics Practice Test

30 Questions – Mixed Conceptual & Calculations