# Introduction

In this activity you will:

- learn about a simple model for an object which is thrown at an angle to the ground moving under the effect of gravity
- note that, in this model, the object is modelled as a particle and so air resistance and rotational effects can be ignored.
- find out how to calculate the velocity of the object at a given time, by considering the horizontal motion and vertical motion separately. Throughout these pages the quantities you see are in metres, seconds or $$\text{ms}^{-1}$$ or $$\text{ms}^{-2}$$ as appropriate.

# Watch it go

The horizontal component of the velocity is constant throughout the motion.

The vertical component is positive when the object is moving up, $$0$$ at the moment the object reaches its highest point and negative after that when the object is dropping.

You will look at how you can calculate velocity at a given time in this model next.

Here you can set the initial velocity of an object and then throw it and watch its velocity over time according to the simple model you'll study here.

'Right' and 'up' are taken to be the positive directions.

Move the green point to set the initial velocity.

Use the **Throw** button to see the motion.

Then press **Set initial velocity** to repeat this.

Make as many observations as you can about the velocity over time.

Finish when you are ready to move on.

Created with GeoGebra

# Initial velocity

In this model you consider the motion in components, the horizontal motion and the vertical motion.

Using trigonometry to calculate the initial vertical and horizontal velocity is crucial for this.

Note the notation used here, $$u_x$$ is the initial horizontal velocity and $$u_y$$ is the initial vertical velocity.

You have seen that, as the initial velocity changes, so does the path taken.

The initial horizontal velocity and the initial vertical velocity are important in the model used here.

Here you are just looking at the initial velocity.

Move the green point to change it.

Look at how the initial horizontal velocity, $$u_x$$, and the initial vertical velocity, $$u_y$$, are calculated using the angle shown.

**Finish** when you want to move on.

Created with GeoGebra

# Calculating velocity

These ideas allow you to calculate values that may be of interest to you such as

- the point in time when the object is at its maximum height
- the time taken to return to ground level.

In this model the only force acting on the object, which is modelled as a particle, is its weight. On Earth, this produces a downwards acceleration of $$9.8\text{ms}^{-2}$$.

In the horizontal direction there is no acceleration and so the horizontal velocity is constant.

The horizontal velocity $$v_x$$ and the vertical velocity $$v_y$$ are calculated at time $$t$$ using $$v=u+at$$.

$$u_x$$ and $$u_y$$ are calculated using the initial speed and angle of projection using trigonometry.

'Up' and 'right' are taken as positive.

Set the initial velocity by moving the green point.

**Throw** the object and watch the calculations of $$v_x$$ and $$v_y$$ update as time changes.

**Finish** when you are ready to move on.

Created with GeoGebra

# A closer look

You should have noticed that the horizontal velocity is constant throughout the motion and that the particle is at its highest point when the vertical component of its velocity is zero.

Now look at this again but focus on how the horizontal and vertical components of velocity (shown in purple in the lower view on the right) change over time.

Set the initial velocity by moving the green point.

**Throw** the object and watch how the 'vertical velocity' and 'horizontal velocity' components change over time.

The velocity is shown as an orange vector.

**Finish** when you are ready to move on.

Created with GeoGebra

# Key points test

You may wish to look at earlier pages in this Walkthrough to help you with your answers.

This model for projectile motion is a very powerful one.

It's used in real world simulations and in computer graphic animations.

Here you will answer some questions about what you've seen.

Assume that the object is modelled as particle, that the only force acting on it is its weight and that $$g=9.8$$.

For each of the two questions on the right tick the correct box.

**Submit** to check your answers.

Created with GeoGebra

# Velocity and speed

You may wish to look at earlier pages in this Walkthrough to help you with your answers.

You may have noticed that the speed of the particle is smallest when it is at the highest point in the motion.

This is because, at this point, the vertical component of its velocity is zero, which means the velocity vector has its shortest length.

Here you will look at the speed of the object during its motion, using the same model.

Speed is just the length of the velocity vector.

It can be calculated using Pythagoras's theorem.

Move the green point to set the initial velocity.

Use the **Throw** button to see the motion.

The object's speed and velocity are shown. Use the **Pause** button to review these.

When is the speed of the object the greatest and the smallest?

Press **Set initial velocity** to repeat this.

**Finish** when you are ready to move on.

Created with GeoGebra

# Finally...

Youâ€™ve now completed this activity. Go and explore the other resources!

In this walkthrough you have learned:

- about a simple model for projectile motion, where objects are modelled as particles and the only force acting is assumed to be the object's weight.
- in this model taking 'up' and 'right' as positive, for an object with initial speed $$u$$ projected at an angle $$\alpha$$ to the horizontal, where the gravitational constant is $$g$$:

the initial horizontal velocity is $$u_x=u\cos\alpha$$

the initial vertical velocity is $$u_y=u\sin\alpha$$;

the horizontal velocity at time $$t\text{ s}$$ is $$v_x=u\cos\alpha$$

the vertical velocity at time $$t\text{ s}$$ is $$v_y=u\sin\alpha - 9.8t$$; - the speed of the object is smallest when the object is at the highest point in its motion.