Parcel sliding down a smooth slope
Difficulty: ★★☆A parcel of mass \(3\text{ kg}\) slides \(3.5\text{ m}\) down a smooth slope inclined at \(20^\circ\) to the horizontal. When it reaches the bottom of the slope it has speed \(8\text{ m s}^{-1}\). Find the speed of the parcel at the top of the slope.
Plate dropped to the floor
Difficulty: ★☆☆A waiter drops a plate and it falls \(1.43\text{ m}\) to the floor, where it smashes. Find the speed of the plate when it hits the floor.
Tennis ball: increase in gravitational potential energy
Difficulty: ★☆☆A tennis ball of mass \(57\text{ g}\) is hit and rises through a height of \(1\text{ m}\). Ignoring air resistance, find the increase in the gravitational potential energy of the ball.
Tennis ball: horizontal speed at the top
Difficulty: ★★☆A tennis ball of mass \(57\text{ g}\) is hit to give it an initial speed of \(180\text{ km h}^{-1}\). It rises through a height of \(1\text{ m}\). Ignoring air resistance, find the horizontal speed of the ball at the top of its flight.
Box sliding down a smooth ramp
Difficulty: ★☆☆A box slides down a smooth ramp. The height of the ramp is \(20\text{ cm}\) and the length of the ramp is \(2.5\text{ m}\). The box starts from rest. What is the speed of the box when it reaches the bottom of the ramp?
Ball launched up a smooth slope
Difficulty: ★★☆A ball is launched up a smooth slope that makes an angle of \(30^\circ\) to the horizontal. The ball travels a distance of \(2.5\text{ m}\) up the slope before coming to instantaneous rest. Find the launch speed of the ball.
Pinball machine ball bearing
Difficulty: ★★☆In a pinball machine, ball bearings are fired up a slope inclined at \(10^\circ\) to the horizontal. After travelling \(1.2\text{ m}\) the ball bearings reach a curved barrier. Find the maximum initial speed of a ball bearing if it comes to rest before getting to the curved barrier.
Football speed at 1 m above the ground
Difficulty: ★★☆A football is kicked from ground level with speed \(15\text{ m s}^{-1}\) and rises to a height of \(1.45\text{ m}\). Assume that air resistance is negligible. Find the speed of the ball when it is \(1\text{ m}\) above the ground.
Football launch angle
Difficulty: ★★★A football is kicked from ground level with speed \(15\text{ m s}^{-1}\) and rises to a maximum height of \(1.45\text{ m}\). Assume that air resistance is negligible. At the top of its flight the ball is travelling horizontally. Show that the ball was kicked at an angle of \(21.0^\circ\) to the horizontal.
Sledge down an icy hill
Difficulty: ★☆☆A boy sits on a sledge at the top of an icy hill. He gently sets the sledge in motion. When he reaches the bottom of the hill he is moving at \(v\text{ m s}^{-1}\). Assuming that friction is negligible, find an expression for the height \(h\) of the hill in terms of \(v\).
Diver from a 10 m board
Difficulty: ★☆☆A diver jumps from a board \(10\text{ m}\) above the water. The diver has initial velocity \(u\text{ m s}^{-1}\) upwards. Find the speed when the diver hits the water.
Energy on a smooth curved track: check 1
Difficulty: ★★☆Info
Maths4U Original · Short practice
A bead of mass 7 kg is constrained to a smooth fixed guide in a vertical plane. At A its speed is \(\displaystyle \frac{6m}{s}\). It descends a vertical height \(\displaystyle \frac{133}{20}\) m to B, then the guide rises far enough for it to reach its first stop. Gravity is the only force doing work, and the bead remains on the guide. Use \(\displaystyle g= \frac{10m}{s^{2}}\).
Find its speed at B in m/s.
Enter one exact number or fraction unless the question specifies rounding. Use the requested units and positive direction; enter the number without its unit. Angles here use degrees. Give full reasoning separately for written problems.
Energy on a smooth curved track: review variant 1
Difficulty: ★★☆Info
Maths4U Original · Short practice
A bead of mass 8 kg is constrained to a smooth fixed guide in a vertical plane. At A its speed is \(\displaystyle \frac{7m}{s}\). It descends a vertical height \(\displaystyle \frac{147}{20}\) m to B, then the guide rises far enough for it to reach its first stop. Gravity is the only force doing work, and the bead remains on the guide. Use \(\displaystyle g= \frac{10m}{s^{2}}\).
Find its speed at B in m/s.
Enter one exact number or fraction unless the question specifies rounding. Use the requested units and positive direction; enter the number without its unit. Angles here use degrees. Give full reasoning separately for written problems.
Energy on a smooth curved track: check 2
Difficulty: ★★☆Info
Maths4U Original · Short practice
A bead of mass 8 kg is constrained to a smooth fixed guide in a vertical plane. At A its speed is \(\displaystyle \frac{7m}{s}\). It descends a vertical height \(\displaystyle \frac{147}{20}\) m to B, then the guide rises far enough for it to reach its first stop. Gravity is the only force doing work, and the bead remains on the guide. Use \(\displaystyle g= \frac{10m}{s^{2}}\).
Find its kinetic-energy increase from A to B in J.
Enter one exact number or fraction unless the question specifies rounding. Use the requested units and positive direction; enter the number without its unit. Angles here use degrees. Give full reasoning separately for written problems.
Energy on a smooth curved track: review variant 2
Difficulty: ★★☆Info
Maths4U Original · Short practice
A bead of mass 9 kg is constrained to a smooth fixed guide in a vertical plane. At A its speed is \(\displaystyle \frac{2m}{s}\). It descends a vertical height \(\displaystyle \frac{221}{20}\) m to B, then the guide rises far enough for it to reach its first stop. Gravity is the only force doing work, and the bead remains on the guide. Use \(\displaystyle g= \frac{10m}{s^{2}}\).
Find its kinetic-energy increase from A to B in J.
Enter one exact number or fraction unless the question specifies rounding. Use the requested units and positive direction; enter the number without its unit. Angles here use degrees. Give full reasoning separately for written problems.
Energy on a smooth curved track: check 3
Difficulty: ★★☆Info
Maths4U Original · Short practice
A bead of mass 9 kg is constrained to a smooth fixed guide in a vertical plane. At A its speed is \(\displaystyle \frac{2m}{s}\). It descends a vertical height \(\displaystyle \frac{221}{20}\) m to B, then the guide rises far enough for it to reach its first stop. Gravity is the only force doing work, and the bead remains on the guide. Use \(\displaystyle g= \frac{10m}{s^{2}}\).
Find how far above A it can rise before its first stop, in m.
Enter one exact number or fraction unless the question specifies rounding. Use the requested units and positive direction; enter the number without its unit. Angles here use degrees. Give full reasoning separately for written problems.
Energy on a smooth curved track: review variant 3
Difficulty: ★★☆Info
Maths4U Original · Short practice
A bead of mass 3 kg is constrained to a smooth fixed guide in a vertical plane. At A its speed is \(\displaystyle \frac{3m}{s}\). It descends a vertical height \(\displaystyle \frac{247}{20}\) m to B, then the guide rises far enough for it to reach its first stop. Gravity is the only force doing work, and the bead remains on the guide. Use \(\displaystyle g= \frac{10m}{s^{2}}\).
Find how far above A it can rise before its first stop, in m.
Enter one exact number or fraction unless the question specifies rounding. Use the requested units and positive direction; enter the number without its unit. Angles here use degrees. Give full reasoning separately for written problems.
Energy on a smooth curved track: check 4
Difficulty: ★★☆Info
Maths4U Original · Short practice
A bead of mass 3 kg is constrained to a smooth fixed guide in a vertical plane. At A its speed is \(\displaystyle \frac{3m}{s}\). It descends a vertical height \(\displaystyle \frac{247}{20}\) m to B, then the guide rises far enough for it to reach its first stop. Gravity is the only force doing work, and the bead remains on the guide. Use \(\displaystyle g= \frac{10m}{s^{2}}\).
Find the greatest vertical height above B in m.
Enter one exact number or fraction unless the question specifies rounding. Use the requested units and positive direction; enter the number without its unit. Angles here use degrees. Give full reasoning separately for written problems.
Energy on a smooth curved track: review variant 4
Difficulty: ★★☆Info
Maths4U Original · Short practice
A bead of mass 4 kg is constrained to a smooth fixed guide in a vertical plane. At A its speed is \(\displaystyle \frac{4m}{s}\). It descends a vertical height \(\displaystyle \frac{273}{20}\) m to B, then the guide rises far enough for it to reach its first stop. Gravity is the only force doing work, and the bead remains on the guide. Use \(\displaystyle g= \frac{10m}{s^{2}}\).
Find the greatest vertical height above B in m.
Enter one exact number or fraction unless the question specifies rounding. Use the requested units and positive direction; enter the number without its unit. Angles here use degrees. Give full reasoning separately for written problems.