Dynamics and Space Learning Intention You will be able to: Explain projectile motion in terms of unbalanced forces. Carry out calculations of projectile motion from a horizontal launch using appropriate relationships and graphs. Investigate and calculate ‘drop time’ and ‘time of flight’ using Area under vH-t graphs for horizontal range, and area under vV-t graphs for vertical height. vH = s / t (constant horizontal velocity) vV = u + at (constant vertical acceleration)

A projectile is any object that is dropped, fired or kicked or otherwise launched in the Earth’s gravitational field. Examples include bullets, arrows, golf balls, water jets etc. To understand how these projectiles travel we study their horizontal and vertical motions at the same time. Let’s look at a jet of water under a stroboscope light in order to study its path.

1(a) What do you observe about the horizontal spacing of the droplets? (b) What does this tell you about the horizontal motion of the water jet?   2(a) What do you observe about the vertical spacing of the droplets? (b) What does this tell you about the vertical motion of the water jet? 3 What force is pulling the water droplets down into their curved path?

The motion of a projectile can be treated as two independent motions: (i) constant velocity in the horizontal direction constant acceleration in the vertical direction due to gravity (= 10 m/s2). When we are solving problems on projectiles, we can treat the motion in the horizontal direction separately from the motion in the vertical direction.

the horizontal motion is a constant speed so we can use d = v t   the vertical motion is a constant acceleration of 10 m/s2. So we can use a=v – u. t eg A ball is thrown horizontally off a cliff at 15 m/s. It lands in the sea 3 s later. Find (a) the horizontal distance from the cliff that it lands the horizontal speed at the time of impact (c) the vertical speed at the time of impact (a) d = v t d= d = 15x3 v = 15m/s d = 45m t = 3 s (b) horizontal speed is constant so horizontal speed on hitting the water = 15m/s. (c) a = v – u u = 0 m/s t v = 10 = v – o a = 10 m/s2 3 t = 3s vertical speed on hitting water = 30 m/s

1 A ball is thrown horizontally off a cliff at 8 m/s 1 A ball is thrown horizontally off a cliff at 8 m/s. It lands in the sea 2.5 s later. (a) Draw a picture of the flight of the ball. (b) Calculate the horizontal distance from the cliff that it lands in the sea. (c) Calculate the horizontal speed at the time of impact with the sea. (d) Calculate the vertical speed at the time of impact with the sea.

2 A ball is thrown horizontally out of a window at 12 m/s 2 A ball is thrown horizontally out of a window at 12 m/s. It lands on the ground 1.2 s later. (a) Draw a picture of the flight of the ball. (b) Calculate the horizontal distance from the window that it lands on the ground. (c) Calculate the horizontal speed at the time of impact with the ground. (d) Calculate the vertical speed at the time of impact with the ground.

3 Sir Isaac Newton once conducted a “thought experiment”. He knew he couldn’t actually do the experiment itself at that time but he imagined what would happen.   (i) Take a large gun to the top of a mountain. Fire the gun. Watch where the shell lands. (ii) Fire a more powerful gun from the same mountain. The shell would land further away. (iii) Can you fire a shell so fast that it would travel right round the world and arrive back at the same mountain? Do you think that it would be possible? You may find it useful to draw a series of pictures to show Newton’s “thought experiment”.

4. A stone thrown horizontally from a cliff lands 24 m out from the cliff after 3 s. Find: a) the horizontal speed of the stone b) the vertical speed at impact.

5. A ball is thrown horizontally from a high window at 6 m/s and reaches the ground after 2 s. Calculate: a) the horizontal distance travelled b) the vertical speed at impact.

6. An aircraft flying horizontally at 150 m/s, drops a bomb which hits the target after 8 s. Find: a) the distance travelled horizontally by the bomb b) the vertical speed of the bomb at impact c) the distance travelled horizontally by the aircraft as the bomb fell d) the position of the aircraft relative to the bomb at impact.

b Calculate the vertical speed of the ball just before it lands. A ball is kicked horizontally off the top of a wall. The initial speed of the ball is 12 m/s. The ball takes 0.9s to land. 12 m/s a Calculate the horizontal distance travelled by the ball before it lands [the range]. b Calculate the vertical speed of the ball just before it lands. c Calculate the height of the wall. a Since the horizontal speed of a projectile remains constant, the range can be calculated using; range = vh x t = 12 x 0.9 = 10.8 m b The ball accelerates downwards at 10 m/s2. The speed can be calculated using: v = u + at speed = a x t = 10 x 0.9 = 9 m/s

c To calculate the height of the wall, a speed time graph of the vertical motion needs to be drawn. The height of the wall will be the vertical distance travelled, which is the same as the area under the graph. area = ½ x b x h = ½ x 0.9 x 12 = 5.4 m.

7. A ball is projected horizontally at 15 m/s from the top of a vertical cliff. It reaches the ground 5 s later. For the period between projection until it hits the ground, draw graphs with numerical values on the scales of the ball’s a) horizontal velocity against time b) vertical velocity against time c) From the graphs calculate the horizontal and vertical distances travelled.

8. In the experimental set-up shown below, the arrow is lined up towards the target. As it is fired, the arrow breaks the circuit supplying the electromagnet, and the target falls downwards from A to B. electromagnet holds target in place a) Explain why the arrow will hit the target. b) Suggest one set of circumstances when the arrow would fail to hit the target (you must assume it is always lined up correctly).

SQA I2 2002 Q22. Table tennis players can practise using a device which fires balls horizontally. The following graphs describe the horizontal and vertical motions of a ball from the instant it leaves the device until it bounces on the table 0.25 s later. The effects of air resistance are assumed to be negligible.

Explain why the shape of the path taken by the ball is curved. (b) (i) What is the instantaneous speed of the ball as it leaves the device? (ii) Describe a method of measuring the instantaneous speed of the ball as it leaves the device. (iii) Calculate the height above the table at which the ball is released. (c) The device is adjusted to fire a second ball which lands at the end of the table. The height and position of the device are not changed. The length of the table is 2.8 m. Assuming that the effects of air resistance are negligible, calculate the instantaneous speed of the second ball as it leaves the device.