# Physics 16 How Far Does a Projectile Go?

When a projectile such as a ball is tossed or thrown, we saw the ball has two big vectors. One is gravity pulling the ball downward. The other is the force pushing the ball forward.

Finding how far a projectile goes is influenced by both of these vectors. But how?

Try this experiment [a friend observing helps]: Stand still and drop a ball catching it on the first bounce. Do not throw it. Ask your friend the path the ball followed down and up.

A dropped ball goes straight down due to gravity then bounces straight up due to elasticity of the ball.

Next walk across the floor. As you walk, drop the ball and catch it on the first bounce. Ask your friend about the path this ball followed.

When you were standing still, the ball had to go straight down and up for you to catch it. When you were walking, if the ball dropped straight down and came straight up, you could not catch it because you were moving. The ball had to follow a curved or projectile pathway.

When you walk forward, you have momentum. Even when you just drop a ball, it has some of the momentum from you so ti follows a curved path down and up so you can catch it instead of passing it by.

This shows the two movements of a projectile, although influenced by each other, work independently. This is important for finding out how far a projectile will go.

Question: How far does a projectile go?

Materials:

Rubber ball

Meter stick

Stopwatch

Procedure:

Repeat dropping and catching the ball as you walk to be sure you are dropping the ball not dribbling it. The only downward force is supposed to be gravity.

Measure the distance from where you will drop the ball to the floor

Mark a starting line on the floor. You will have to be walking as you cross this line so give yourself some room.

The meter stick will probably be long enough to show how far you walk dropping and catching the ball. Set is up from your starting line along the path you will follow.

Put the meter stick down with one end touching the starting line

Holding the ball, start walking toward the starting line

As you cross the starting line, drop the ball and start the stopwatch

Stop, stop the stopwatch when you catch the ball on the first bounce

Find the distance you went by looking at the meter stick

[You may want to have your friend work the stopwatch.]

Doing this can be difficult so you may want to repeat it a few times

Observations:

Describe the path the ball follows when you drop it:

Describe the path the ball follows when you drop it while walking:

Distance the ball is dropped:

For each trial with the ball write down the time and distance:

Analysis:

The falling projectile ball has two different movements so there are two different vectors.

When you drop a ball while walking, ti follows a curved path. In vectors this is shown with two vectors, one going forward for the momentum and one going down for the gravity. Added up the two show where the ball will land.

Vector 1 – Distance

One of the vectors points forward because you are moving forward. How far forward will the ball go?

We’ve seen this already. The distance is the time multiplied by the speed. You want to know about the part when the ball is going to the floor. As we saw in the last Project, this is half of the total. Divide the distance and the time in half.

Now you can rearrange the formula to v = d/t. Put in your values for d and t to find the velocity for the first vector.

Vector 2 – Falling

We’ve seen this already. Gravity is acceleration so the distance down equals gravity’s acceleration multiplied by the square of the time.

Remember the acceleration is 9.8 m/s2 and the time is what you used before. Put your values in and calculate the distance the ball dropped.

Now compare the distance you dropped the ball to what you calculated.

Looking at What Makes Your Values Change

We used several measurements in our calculations. One was the value for gravity. We used an accepted and proven value of 9.8 m/s2 so this will not change.

There are three measurements you need for a projectile. One is time. One is the distance to the ground. The third is the distance from dropping to catching the ball.

Now we have several measurements we made. One was the distance you walked between dropping and catching the ball. We assumed we dropped and caught the ball at the same height.

How accurately did you measure the distance? It must be in meters. If you measured the distance and the time in different tests, can you be sure the distance was the same both times?

Another distance was how far you dropped the ball. When should we have measured this distance? Why? Is there a way to make sure you drop the ball from the same height each time?

We also measured the time it took to bounce the ball. My ball bounced up very fast making it hard to time this accurately. What about your time?

In the last Project we threw a ball upwards. The force we threw it with was countered by gravity until the two were equal and gravity took over causing the ball to fall down. Half the time was spent going up and half coming down.

When we drop a ball, gravity pulls it down. When the ball bounces, the force of the ball hitting the ground makes it come back up. Different kinds of balls will bounce differently.

What kind of ball did you use? I used one of those rubber balls that bounces back as high as the height I dropped it from for the first bounce. This kind of ball would come very close to taking the same amount of time coming up as it did going down.

If you used a different kind of ball, this may not be true. That would mean your time value would not be right. That would make your calculations off too.

Taking Another Look At How This Works

A projectile differs from a falling object because it moves from a starting to an ending place. The projectile or ball we looked at went out horizontally and fell as it traveled across this distance.

The arc of the falling projectile has two vector parts. One is falling because of gravity. The other is going forward due to some force pushing it forward.

With the balls we dropped, what was the forward push? When we walk forward, we have something called momentum. We will look at this more closely in a future Project. This momentum provided the force forward to our balls.

Calculating the time it takes for the ball to fall to the ground uses the formula d = at2 where a is 9.8 m/s2 for gravity and d is the distance we measure.

Calculating how far forward the ball will go uses the formula d = vt. If we know how far the ball will fall, we can calculate time as above or we can measure it.

We must know the distance to calculate velocity or velocity to calculate distance. We must know two values for the formula to calculate the third.

Giving It another Try

This time let’s try to keep our measurements much more accurate. The first one is how far we will drop the ball.

The farther the ball drops, the longer it will take making timing easier. However, we need to have some mark so we drop it and catch it at the same height. The waist is convenient but may make timing difficult.

Measure the height to your waist.

Next is the distance you walk while bouncing the ball.You need to be walking along before you drop the ball. Have a route and a marked starting line. It may help to have a line or something marked to keep walking in a straight line. Have a way to mark the ending spot.

The hardest measurement is the time. The ball will bounce back up quickly. Having a friend time the bounce may be easier than trying to walk a straight line, drop the ball while starting the stopwatch and catching the ball as you stop the stopwatch.

You can do several trials but keep each set of measurements separate as each trial may differ in  measurements from the others. They should be similar so you can pick a set that seems more accurate to use for your calculations.

What Can You Calculate?

You can use your distance and time measurements to calculate your walking velocity using v = d/t.

You can use d = at2 where a = 9.8 m/s2 to calculate the distance the ball falls to the ground or the time it takes to reach the ground. Because the time is squared, you will want to use a calculator to find the square root or the time that was squared.

What We Will Do Next

Next week the Project will be a projectile challenge. We will calculate the velocity of a ball traveling off a table and use it to calculate where to place a cup to catch the ball when it reaches the ground. You will need two long ramps, a ball, a meter stick and a stopwatch. Oh, you will need a cup to catch the ball in.

# Physics 15 Projectile Motion

There’s straight line motion. Our balls and cars have shown us a little about it.

There’s pendulum motion. Nuts on strings showed us about this.

There’s circular motion. We used a nut on a string to find out a little about it.

One last type of motion is projectile motion.

What happens if you throw a ball straight up?

If you don’t dodge, it will come straight down and hit you. Why?

If you throw a ball across a room, it curves down to the floor. Why?

Question: How does projectile motion work?

Materials:

Ball

Stopwatch

Paper

Pencil

Procedure:

Toss the ball up from your hand

The ball leaves the hand, goes up then comes back down into the hand.

Observe how the ball goes up and down

Catch it when it returns to your hand

If you have a friend to help, have your friend gently toss the ball across a space

Observe how the ball moves

Throwing a ball or other projectile gives it an arch shaped path.

Play catch outside with your friend

Start with gentle tosses and gradually throw the ball harder

Observe how the motion of the ball changes as you throw it harder

Start the stopwatch and throw your ball straight up as hard as you can [It helps to have a friend help with this.]

Stop the stopwatch when the ball hits the ground

Repeat this only if you did not get the stopwatch stopped on time

Observations:

Draw your ball going up and down one time

Describe how your ball goes up and down

Draw your ball going across a space

Describe the motion of your ball

Describe how the motion of your ball changes as the throws get harder

Time for your ball to go up and down:

Conclusions:

What makes your ball go up?

Newton’s First Law of Motion says an object in motion will continue that motion unless acted on by another force. What force keeps your ball from going up forever?

Why do you think the projectile motion of your ball changes as you change how you throw it?

Throwing a ball is more like the projectile motion people think of because the ball goes over a distance. At the beginning most of the force pushes the ball up, some goes sideways and gravity pulls down. At the top of the arch there is no more force pushing the ball up but it still has force pushing it sideways and gravity pulls it down. When the ball lands, only gravity is still pulling on the ball.

Draw your ball going across a space. Show your ball at the beginning, middle and end of the toss.

Add vectors to show how the forces are acting on your ball at each point to change how it moves.

How does projectile motion work?

Why can’t you use an average time for throwing your ball straight up?

Analysis:

How high did you throw your ball?

Your ball spent half its time going up and half its time coming down. Divide your time in half.

What provided the force to make the ball go up?

When the ball first leaves the hand, most of the force is pushing the ball upward with gravity pulling against it. At the top of the loop, gravity and upward force cancel each other out and the ball stops. Then gravity pulls harder than the upward force so the ball falls back down into your hand. This is projectile motion.

What provided the force to make the ball come down?

Remember the formula from the last Project was a = d/t2

This time we know the acceleration is 9.8 m/s2 and the time and want to know the distance. We can rearrange the formula to be d = at2

Calculate how high you threw the ball.

What I Found Out

First I found out this Project is much easier with two people and I am only one so pictures were not possible. So I did drawings on my computer.

Tossing a ball up and down in one hand isn’t hard. The ball went up out of my hand then stopped and fell back down into my hand.

Playing catch can be fun. When the ball is tossed easily, it arches up then down into the other person’s hands. As the ball is tossed harder, the arch flattens out until it is almost a straight line.

Throwing or tossing a ball uses force from my hand. Gravity is always pulling down on the ball.

Throwing a ball harder means it is going faster so gravity doesn’t slow it down as fast flattening the curve the ball makes.

The ball must go fast enough to overcome gravity. The harder I throw the ball, the faster it goes and the longer before gravity slows it down enough to make it fall.

In projectile motion the ball starts off with lots of force pushing it upwards. Gravity pulls a little of that force at a time slowing the ball down. At the top of the arch gravity is equal to the force making the ball go forward. Then gravity is greater making the ball fall down.

When a ball goes straight up, gravity pulls down until the ball stops and starts to fall down. Even if I try very hard, I won’t throw the ball with the same force every time so I must time each throw separately.

When I tossed my ball up, it took 1.94 sec to hit the ground. Half the time is .97 sec. Squaring the time gives .88 s2. Multiplying that by 9.8 m/s2 tells me I threw the ball 8.6 m or about 28 feet up.