Mechanics — practice questions

A cyclist travels $450$ m along a straight road in $30$ s.

Calculate the cyclist's average speed.

A swimmer swims $50$ m north up a pool in $40$ s, then $50$ m back south in $45$ s.

Calculate her average speed and her average velocity for the whole swim, and explain why the two answers are different.

A ball is thrown vertically upward. At the highest point of its flight, a student claims that because the ball is momentarily stationary, its acceleration must also be zero.

Discuss whether the student is correct. In your answer you should refer to the definitions of velocity and acceleration, and to the force acting on the ball.

A car starts from rest and accelerates uniformly at $3.0$ m s⁻² for $6.0$ s.

Calculate its final velocity.

A motorcycle travelling at $24$ m s⁻¹ brakes uniformly and stops in $4.0$ s.

Calculate the acceleration and the distance travelled while braking. Explain why the acceleration is negative.

Two cars, A and B, are side by side and both travelling at $20$ m s⁻¹ when they begin to brake. Car A decelerates uniformly at $5.0$ m s⁻²; car B decelerates uniformly at $2.5$ m s⁻².

Calculate each stopping distance. Then explain fully, using the equations of motion, why halving the deceleration doubles the stopping distance, and why doubling the initial speed instead would have a far larger effect.

A ball is dropped from rest from a height of $20$ m. Take $g = 9.8$ m s⁻².

Calculate the speed at which it hits the ground.

A stone is thrown vertically upward at $14.7$ m s⁻¹.

Calculate the maximum height it reaches above the throwing point, and explain why the stone's acceleration is not zero at the highest point of its flight.

A physics class drops a cricket ball and a table-tennis ball of the same diameter from a first-floor window at the same instant. The cricket ball lands noticeably first.

Explain fully why the free-fall model predicts that they should land together, and why in reality they do not. Refer to the forces on each ball and to Newton's second law.

A distance–time graph is a straight line passing through $(0, 0)$ and $(8.0\text{ s}, 24\text{ m})$.

Calculate the velocity of the object.

A distance–time graph for a bus consists of a straight line rising from $(0, 0)$ to $(30\text{ s}, 300\text{ m})$, followed by a horizontal line to $(50\text{ s}, 300\text{ m})$.

Describe the motion of the bus in each stage and calculate the velocity in each. Explain what feature of the graph tells you the bus is stationary.

Two distance–time graphs are drawn on the same axes for the same $10$ s interval. Graph P is a straight line from $(0, 0)$ to $(10\text{ s}, 50\text{ m})$. Graph Q is a curve from $(0, 0)$ to $(10\text{ s}, 50\text{ m})$ that starts nearly flat and finishes very steep.

Compare the motion shown by the two graphs. Your answer should discuss average velocity, instantaneous velocity, and acceleration, and explain how the same average velocity can arise from completely different motions.

A car's velocity–time graph is a horizontal line at $12$ m s⁻¹ for $8.0$ s.

State what the area under the line represents, and calculate the distance the car travels.

A cyclist's velocity–time graph rises in a straight line from $0$ to $10$ m s⁻¹ over $4.0$ s, then stays at $10$ m s⁻¹ for a further $6.0$ s.

Calculate the acceleration during the first stage and the total distance travelled, and explain how you know the acceleration in the second stage is zero.

A lift's velocity–time graph shows: a straight rise from $0$ to $2.0$ m s⁻¹ over $2.0$ s; a horizontal section at $2.0$ m s⁻¹ for $6.0$ s; then a straight fall back to $0$ over $4.0$ s.

Analyse the motion fully. Calculate the acceleration in each stage and the total distance, and explain how the graph shows that the passenger feels heaviest in the first stage and lightest in the last.

A ball is dropped from a height of $5.0$ m. Take $g = 9.8$ m s⁻².

Calculate the time it takes to reach the ground.

A ball is kicked horizontally at $6.0$ m s⁻¹ from the top of a wall $1.8$ m high.

Calculate how far from the base of the wall it lands, and explain why its horizontal velocity does not change during the flight.

A ball is thrown horizontally from a balcony at the same instant that an identical ball is dropped from the same height.

Explain fully why the two balls land at the same time, and describe how the landing speeds compare. Support your answer with reference to the components of velocity.

A ball is launched at $25$ m s⁻¹ at an angle of $40°$ above the horizontal.

Calculate the horizontal and vertical components of the launch velocity.

A golf ball is hit from level ground at $30$ m s⁻¹ at $35°$ above the horizontal.

Calculate the maximum height it reaches, and explain why the horizontal component of its velocity is the same at the top of the flight as it was at launch.

A shot-putter can release the shot at a fixed speed. She is told that launching at $30°$ and at $60°$ would send the shot the same horizontal distance on level ground.

Explain fully why the two angles give the same range, using the components of the launch velocity, and discuss which of the two would be the better choice in a real competition.

A ball on a string moves in a horizontal circle of radius $0.80$ m at a constant speed of $4.0$ m s⁻¹.

Calculate its centripetal acceleration.

A stone on a string is whirled in a horizontal circle of radius $1.2$ m, completing $2.5$ revolutions per second.

Calculate its speed and centripetal acceleration, and explain why the stone is accelerating even though its speed is constant.

A ball is being whirled in a horizontal circle on the end of a string when the string suddenly breaks.

Describe and explain the subsequent path of the ball. Your answer should refer to Newton's laws and to why passengers in a cornering car feel pushed outward, even though no outward force acts on them.

A student has a mass of $58$ kg.

Calculate their weight on Earth, where $g = 9.8$ m s⁻².

A $6.0$ kg box is pushed along a horizontal bench with a force of $30$ N. Friction of $12$ N opposes the motion.

Calculate the net force on the box, and explain why the normal force from the bench is equal to the box's weight in this situation.

A skydiver jumps from a plane. She accelerates downward at first, but after some time she falls at a constant velocity known as terminal velocity.

Using free-body diagrams and Newton's laws, explain fully how the forces on her change from the moment she jumps until she reaches terminal velocity, and explain what happens to those forces immediately after she opens her parachute.

Two forces act on an object: $12$ N to the right and $5.0$ N to the left.

Calculate the net force.

A yacht is acted on by a $900$ N force from the wind pushing it east and a $1200$ N force from the current pushing it south.

Calculate the size and direction of the resultant force, and explain why the resultant is not simply $2100$ N.

Two tugboats tow a ship. Each pulls with a force of $8.0$ kN, and the two cables make an angle of $60°$ with each other, symmetrically about the ship's forward direction.

Using a scale diagram or trigonometry, determine the resultant force on the ship. Then explain fully how the resultant would change if the angle between the cables were increased toward $180°$, and why tug crews try to keep the angle small.

A force of $50$ N acts at $30°$ above the horizontal.

Calculate its horizontal component.

A $40$ kg crate rests on a ramp inclined at $15°$ to the horizontal.

Calculate the component of the crate's weight acting down the slope and the normal force from the ramp. Explain why the normal force is less than the crate's weight.

A student must move a heavy suitcase across a floor. She can either push it with a force of $200$ N directed $30°$ below the horizontal, or pull it with the same $200$ N force at $30°$ above the horizontal. The suitcase has a mass of $30$ kg.

Analyse both options. Calculate the horizontal driving force and the normal force in each case, and explain fully which method makes the suitcase easier to move.

A net force of $36$ N acts on an object of mass $9.0$ kg.

Calculate the acceleration of the object.

A $1200$ kg car experiences a driving force of $3600$ N and resistive forces totalling $1200$ N.

Calculate the car's acceleration. Then explain what happens to the acceleration as the car speeds up, if the driving force stays constant.

A student says: "Newton's third law says that when a horse pulls a cart, the cart pulls back on the horse with an equal force. The two forces are equal and opposite, so they must cancel out and the cart can never move."

Explain fully why the cart does in fact accelerate. Your answer should identify the third-law pairs involved and explain the difference between a third-law pair and balanced forces.

A $0.30$ kg ball on a string moves in a horizontal circle of radius $1.2$ m at $4.0$ m s⁻¹.

Calculate the centripetal force acting on the ball.

A $1200$ kg car rounds a bend of radius $40$ m at a constant $15$ m s⁻¹.

Calculate the centripetal force required, name the force that provides it, and explain why the car is accelerating even though its speed is constant.

A driver takes a corner of radius $50$ m at $54$ km h⁻¹ in the dry with no difficulty. The following week, in wet conditions on the same corner at the same speed, the car slides toward the outside of the bend and off the road.

The car has a mass of $1100$ kg. Calculate the centripetal force required, then explain fully why the car left the road, referring to Newton's laws. Explain also why a passenger felt thrown toward the outside of the car.

A force of $25$ N is applied at right angles to a door handle $0.80$ m from the hinges.

Calculate the torque about the hinges.

A student cannot undo a tight bolt using a $0.20$ m spanner with a force of $180$ N. She fits a $0.60$ m pipe over the handle and applies the same force at the end of it.

Calculate the torque in each case, and explain why the longer lever succeeds.

A gate is $1.8$ m wide and hinged on its left edge. A child pushes with a force of $60$ N at the outer edge, but pushes at $40°$ to the plane of the gate rather than perpendicular to it.

Calculate the torque produced. Explain fully why this is less than the torque from a perpendicular push of the same size, and explain why pushing a gate near its hinges is ineffective no matter how hard you push.

A seesaw is pivoted at its centre. A force of $300$ N acts downward $1.2$ m to the left of the pivot.

Calculate the torque this force produces about the pivot, and state its direction.

A uniform metre rule is pivoted at the $50$ cm mark. A $2.0$ N weight hangs at the $20$ cm mark.

Calculate where a $3.0$ N weight must hang to balance the rule, and explain why the weight of the rule itself can be ignored in this calculation.

A uniform diving board of mass $30$ kg and length $3.0$ m is bolted to the ground at its left end (support A) and rests on a roller support (support B) $1.0$ m from that end. The rest of the board projects over the water. A $55$ kg diver stands at the free right-hand end.

Calculate the forces at supports A and B. Explain fully what the sign of the force at A means physically, and why the bolt at A is necessary.

A spring is stretched by $0.15$ m when a force of $45$ N is applied.

Calculate the spring constant.

A spring has an unstretched length of $20$ cm. A $1.5$ kg mass hung from it stretches it to $26$ cm.

Calculate the spring constant and the elastic potential energy stored at that extension.

A student loads a spring with increasing masses and plots force against extension. The graph is a straight line through the origin up to a force of $40$ N and an extension of $0.16$ m, after which it curves upward less steeply. When the student removes all the loads after reaching $60$ N, the spring no longer returns to its original length.

Explain fully what the graph shows. Determine the spring constant, explain what has happened beyond $40$ N, and explain how the energy stored at $0.16$ m compares with the energy stored at $0.08$ m.

A $60$ kg runner moves at $7.5$ m s⁻¹.

Calculate the runner's momentum.

A $0.058$ kg tennis ball travelling at $25$ m s⁻¹ is struck by a racquet and returns along the same line at $32$ m s⁻¹.

Calculate the change in momentum of the ball, and explain why the change is larger than the momentum the ball had before being struck.

Two identical $0.20$ kg trolleys approach each other along a track, each travelling at $3.0$ m s⁻¹.

Calculate the total momentum and the total kinetic energy of the system. Then explain fully why the total momentum can be zero while the kinetic energy is not, and what this means for what happens when they collide head-on and stick together.

A force of $25$ N acts on an object for $4.0$ s.

Calculate the impulse delivered to the object.

A $0.16$ kg cricket ball travelling at $30$ m s⁻¹ is caught and brought to rest in $0.12$ s.

Calculate the average force exerted on the ball, and explain why a fielder moves their hands backward while catching.

A $70$ kg stunt performer falls from a height and lands at $9.0$ m s⁻¹. In one take she lands on a crash mat and stops in $0.50$ s; in another she lands on bare concrete and stops in $0.040$ s.

Calculate the average force in each case. Explain fully, using impulse, why the crash mat protects her, and explain why bending the knees on landing has the same effect. Include a comment on why a bouncing landing would be worse than one where she comes to rest.

A $2.0$ kg trolley moving at $3.0$ m s⁻¹ collides with a stationary $4.0$ kg trolley and they stick together.

Calculate their common velocity after the collision.

A $0.50$ kg ball moving east at $6.0$ m s⁻¹ collides with a $1.5$ kg ball moving west at $2.0$ m s⁻¹. They stick together.

Calculate their velocity after the collision, and state with a reason whether the collision is elastic or inelastic.

A $6000$ kg railway wagon rolling at $3.0$ m s⁻¹ couples with a stationary $2000$ kg wagon.

Calculate the velocity after coupling and the kinetic energy before and after. Explain fully why momentum is conserved but kinetic energy is not, referring to Newton's third law and to where the energy goes.

A $50$ kg skater standing at rest on ice pushes a $2.0$ kg ball away at $6.0$ m s⁻¹.

Calculate the skater's recoil velocity.

A $700$ kg cannon fires a $8.0$ kg shell horizontally at $210$ m s⁻¹.

Calculate the recoil velocity of the cannon, and explain why the cannon recoils much more slowly than the shell travels.

Two ice skaters, of mass $80$ kg and $50$ kg, stand at rest facing each other and push apart. The lighter skater moves off at $2.4$ m s⁻¹.

Calculate the heavier skater's velocity and the total kinetic energy after the push. Explain fully why the total momentum is zero both before and after, while the kinetic energy increases from zero, and state where that energy came from.

A force of $75$ N pushes a box $4.0$ m across a floor in the direction of the force.

Calculate the work done.

A $65$ kg student runs up a flight of stairs $4.2$ m high in $5.5$ s.

Calculate the work done against gravity and the student's power output. Explain why a second student who climbs the same stairs in $11$ s does the same work but has half the power.

A $1500$ kg car travels at a constant $25$ m s⁻¹ along a level road against total resistive forces of $600$ N. It then climbs a hill that rises $1.0$ m for every $20$ m travelled along the road, maintaining the same speed and facing the same resistive forces.

Calculate the engine power required in each case. Explain fully why more power is needed on the hill even though the speed is unchanged, and explain why the car has zero acceleration in both cases despite the engine working hard.

A $2.0$ kg object moves at $6.0$ m s⁻¹.

Calculate its kinetic energy.

A $0.20$ kg ball is dropped from a height of $3.0$ m and rebounds to a height of $1.8$ m.

Calculate the energy transformed into heat and sound during the bounce, and explain why the ball does not return to its original height.

A $60$ kg skateboarder starts from rest at the top of a ramp $3.5$ m high. She reaches the bottom at $7.2$ m s⁻¹.

Calculate the energy transformed by friction and air resistance, and the speed she would have reached without them. Explain fully why the energy method gives the final speed without any knowledge of the ramp's shape, and explain why the equations of motion could not be used here.