Chapter 15: Quantities, units and kinematics

The chapter problem: does the free kick go over the crossbar?

A free kick is taken when the ball is on the ground. The ball starts $40$ m from the goal line, straight in front of the goal. The player strikes the ball so that it leaves the ground at $20$ m/s at $45^\circ$ to the horizontal.

The crossbar is $2.44$ m above the ground. Model the ball as a particle that moves under gravity alone. Ignore air resistance. Take $g = 9.8$ m/s$^2$.

By the end of the chapter you will work out how high the ball is when it reaches the goal line. Then you will decide whether it passes under the crossbar or over it.

First contact — no working needed yet. Sketch the path of the ball. Mark the goal line and a line for the crossbar. Predict whether the ball passes under the bar or over it. Write one sentence to give your prediction, and one reason for it.

By the end of this chapter you will be able to

• name the fundamental SI quantities and their units, and build derived units from them;
• convert between units, including km/h and m/s;
• use the language of kinematics: position, displacement, distance travelled, speed, velocity and acceleration;
• read the gradient of a displacement–time graph and the gradient and area of a velocity–time graph;
• choose and use the constant-acceleration formulae;
• use calculus to move between displacement, velocity and acceleration;
• model motion under gravity in a vertical plane;
• resolve a velocity into components and solve a projectile problem with those components.

Where it is used

Mechanics starts with measurement. Before you can describe a moving object, you need to say what you are measuring and in what unit. Velocity and acceleration are built from length and time, and force is built from mass as well. The same language then describes a car on a road, a lift in a shaft, a ball in flight and a planet in orbit. Projectile motion is the first model that puts a horizontal and a vertical motion together. The same idea returns in Chapter 16, where forces cause the acceleration, and in the study of physics.

Your turn to connect it

Think of something you have seen move in two directions at once. It could be a ball thrown across a garden, water from a hose, a gymnast in a jump or a parcel dropped from a moving aircraft. Write down the two directions, and say which one is steady and which one changes. Keep your answer for the Review.

Roadmap

1. 15.1 Fundamental and derived SI quantities and units
2. 15.2 The language of kinematics
3. 15.3 Displacement–time and velocity–time graphs
4. 15.4 The constant-acceleration formulae
5. 15.5 Calculus in kinematics
6. 15.6 Motion under gravity and projectiles using vectors
7. Review