A 0.02 kg dart rests against the compressed spring of a toy launcher on a level bench. The spring, constant 288 N/m, is compressed 0.05 m.
How fast does the dart leave the launcher?
Well, count the spring's store first: with k = 288 N/m and x = 0.05 m, the stored energy comes to 0.36 J.
That number is the move you met earlier in this unit, EPE = ½kx², reused: (0.5 × 288) × (0.05 × 0.05) is 0.36 J.
Run the gatekeeper scan: a level launch, friction negligible in the barrel, no collision. Conserved. So every stored joule becomes kinetic: ½mv² must equal 0.36 J.
Unpack the speed: double the 0.36 J and divide by the 0.02 kg mass, so v² = 0.72 / 0.02 = 36, and v = 6 m/s.
A launcher spring, k = 288 N/m, is compressed 0.05 m behind a 0.02 kg dart, friction negligible. Calculate the launch speed.
Write down the values:
k = 288 N/m, x = 0.05 m, m = 0.02 kg, v = ?
Write down the equation:
½kx² = ½mv²
The spring's store: (0.5 × 288) × (0.05 × 0.05), which is 0.36 J.
Substitute in the values, and calculate:
v² = 0.72 / 0.02 = 36
The dart leaves at 6 m/s.
A launcher spring holds 1.6 J of elastic potential energy behind a 0.05 kg ball, friction negligible, level launch. Enter the ball's launch speed in m/s.
Twice the compression stores four times the energy, so the launch speed doubles: the x² from the spring lessons carries straight through.
The same launcher is compressed 0.1 m, twice as far, behind the same dart. Calculate the new launch speed.
Write down the values:
k = 288 N/m, x = 0.1 m, m = 0.02 kg, v = ?
Write down the equation:
½kx² = ½mv²
The spring's store: (0.5 × 288) × (0.1 × 0.1), which is 1.44 J, four times the smaller store.
Substitute in the values, and calculate:
v² = 2.88 / 0.02 = 144
The dart leaves at 12 m/s: double the compression, double the speed.
A spring of constant 500 N/m is compressed 0.2 m behind a 0.2 kg ball, friction negligible, level launch. Enter the ball's launch speed in m/s.
A spring of constant 120 N/m is compressed 0.2 m behind a 0.012 kg dart, friction negligible, level launch. Enter the dart's launch speed in m/s.
A ping-pong-ball launcher uses a spring of constant 400 N/m, compressed 0.15 m behind a 0.04 kg ball, friction negligible, level launch. Enter the ball's launch speed in m/s.
A bench-mounted launcher fires a 0.05 kg dart with a spring of constant 980 N/m compressed 0.1 m, friction negligible, level launch. Enter the dart's launch speed in m/s.
A playground game launches a 0.5 kg ball across level, friction-negligible turf using a plunger spring of constant 800 N/m compressed 0.1 m. Enter the ball's launch speed in m/s.