Work & Kinetic Energy
A force that is always perpendicular to an object’s displacement does this much work on it.
What is zero?
(W = Fd cos 90° = 0. Centripetal force and the normal force on a flat surface are examples.)
In the absence of nonconservative forces, this quantity, the sum of kinetic and potential energy, stays constant.
What is mechanical energy?
A 1500 kg car traveling at 20 m/s has this much momentum.
What is 30,000 kg·m/s?
In this type of collision, kinetic energy is conserved.
What is an elastic collision?
A wheel spinning at 60 rpm has this angular speed in rad/s.
What is 2π ≈ 6.28 rad/s?
(60 rev/min × 2π rad/rev ÷ 60 s/min)
A 10 N force pulls a box 4 m along a floor at an angle of 60° above the horizontal. This is the work done by the force.
What is 20 J?
(W = Fd cos θ = 10 · 4 · 0.5)
A ball is dropped from rest from a height of 10 m. Ignoring air resistance, this is its speed just before it hits the ground.
What is 14 m/s?
(v = √(2gh) = √196)
A 0.15 kg baseball moving at 20 m/s toward the batter is hit straight back at 30 m/s. This is the magnitude of the impulse the bat delivers.
What is 7.5 N·s?
(Δp = 0.15(30 − (−20)))
A 2 kg cart at 6 m/s collides with a stationary 4 kg cart and they stick together. This is their final speed.
What is 2 m/s?
(2(6) = 6v)
A wheel starting from rest has a constant angular acceleration of 4 rad/s². After 5 s, these are its angular speed and angular displacement.
What are 20 rad/s and 50 rad?
(ω = αt and θ = ½αt²)
A 2 kg object speeds up from 3 m/s to 5 m/s. This is the net work done on it.
What is 16 J?
(Wnet = ΔK = ½(2)(5² − 3²))
A 0.5 kg block is pressed against a spring (k = 200 N/m), compressing it 0.20 m, then released on a frictionless horizontal surface. This is the block’s speed when it leaves the spring.
What is 4 m/s?
(½kx² = 4 J = ½(0.5)v²)
A 60 kg skater at rest on frictionless ice throws a 3 kg ball at 10 m/s. This is the skater’s recoil speed.
What is 0.5 m/s?
(0 = 3(10) + 60v)
A 2 kg cart at 6 m/s hits a stationary 4 kg cart in a head-on elastic collision. These are the final velocities of the two carts.
What are −2 m/s (the 2 kg cart bounces back) and +4 m/s (the 4 kg cart)?
(Use v₁ = ((m₁ − m₂)/(m₁ + m₂))v₀ and v₂ = (2m₁/(m₁ + m₂))v₀. Check: Ki = 36 J, Kf = 4 + 32 = 36 J.)
Two 2 kg point masses are at the ends of a 1 m massless rod that rotates about its center at 4 rad/s. This is the system’s rotational kinetic energy.
What is 8 J?
(I = 2(2)(0.5)² = 1 kg·m², so K = ½(1)(4²))
A spring with k = 200 N/m is slowly compressed 0.30 m from its relaxed length. This is the work you do on the spring.
What is 9 J?
(W = ½kx² = ½(200)(0.09))
A 2 kg block slides from rest down a rough ramp from a height of 3 m and reaches the bottom at 6 m/s. This is the energy converted to thermal energy by friction.
What is 22.8 J?
(mgh − ½mv² = 58.8 − 36)
A 6 kg object at rest explodes into two pieces. The 2 kg piece flies east at 12 m/s. This is the speed and direction of the other piece.
What is 6 m/s to the west?
(0 = 2(12) + 4v)
A 10 g bullet embeds in a 1.99 kg block hanging from strings, and the block swings up 0.20 m. This is the bullet’s speed before impact.
What is 396 m/s (≈ 400 m/s)?
(Swing: v = √(2gh) = 1.98 m/s just after impact. Collision: 0.01·vb = 2.00(1.98).)
A force F = (10 î + 10 ĵ) N acts at position r = (0.5 î) m from the axis. This is the torque (using the cross product), and the angular acceleration it produces on an object with I = 2.5 kg·m².
What is τ = 5 k̂ N·m and α = 2 rad/s²?
(τ = r × F = (0.5)(10) k̂ and α = τ/I)
A 4 kg block at rest on a frictionless surface is pushed by a force F = (4x) N, where x is in meters. After the block moves from x = 0 to x = 2 m, this is its speed.
What is 2 m/s?
(W = ∫4x dx = 2x² = 8 J, and ½(4)v² = 8 gives v = 2 m/s)
A 0.2 kg ball on a 0.50 m string swings in a vertical circle. This is the minimum speed it must have at the bottom to complete the circle with the string staying taut.
What is 4.95 m/s?
(At the top, mg = mv²/R gives vtop² = gR. Then ½mvbottom² = ½mvtop² + mg(2R), so vbottom = √(5gR) = √24.5. The mass doesn’t matter.)
A stationary 3 kg firework explodes into three 1 kg pieces. One moves east at 6 m/s and another moves north at 8 m/s. This is the velocity of the third piece, and the total kinetic energy released.
What is 10 m/s at 53° south of west, with 100 J released?
(px = −6 and py = −8 for the third piece, so |v| = 10 m/s and θ = tan⁻¹(8/6). K = ½(36 + 64 + 100) = 100 J.)
A 2 kg ball moving at 5 m/s has a glancing elastic collision with an identical stationary ball. The first ball is deflected 30° from its original path. These are the speeds of both balls afterward.
What are 4.33 m/s (first ball) and 2.5 m/s (second ball, moving 60° on the other side)?
(For equal masses in an elastic collision with one at rest, the balls leave at 90° to each other. v₁ = 5 cos 30° and v₂ = 5 sin 30°.)
A solid cylinder rolls without slipping from rest down an incline, dropping a vertical height of 3 m. This is its speed at the bottom.
What is about 6.26 m/s?
(mgh = ½mv² + ½Iω² with I = ½MR² and ω = v/R, so mgh = ¾mv². Then v = √(4gh/3) = √39.2.)