A fielder standing near the boundary lets a ball slip from his hand. It falls straight down. A moment later he throws another ball to the wicket-keeper. That ball rises, turns over and comes down in a long curve. Both balls feel the same gravity, the same \(10\,\text{m/s}^2\) straight down. One path is a straight line and the other is a curve.
Two arrows decide the shape of a path
Motion in Two Dimensions gave two facts about a moving body. Its velocity always lies along the tangent to its path. Its acceleration is the rate at which that velocity changes, in size or in direction. This chapter is built on one question: when does the direction of the velocity change?
Suppose the acceleration points exactly along the velocity. In a short time it adds a small velocity in the same direction. The velocity grows, but its direction stays the same. If the acceleration points exactly against the velocity, it removes a little velocity each instant. The body slows down, and again its direction does not change. In both cases the body stays on one straight line.
Now suppose the acceleration makes an angle with the velocity. In a short time it adds a small velocity that points partly sideways. The new velocity is the old one plus this sideways piece, so it points in a new direction. The tangent to the path has turned, and the path bends. It bends towards the side on which the acceleration points.
A body moves along a straight line only when its acceleration is zero or lies along the line of its velocity. If the acceleration has any part across the velocity, the direction of motion changes and the path bends towards that part.
The fielder's two balls fit this rule. The dropped ball starts with its velocity downward, and gravity also acts downward. The two arrows lie along one line, so the path is straight. The thrown ball starts with its velocity forward and upward, while gravity acts straight down. The arrows make an angle, so the path bends downward, towards gravity.
A ball is thrown vertically upward. Air resistance is neglected. What is the shape of its path?
Two parts of the acceleration
At any point of a path, draw two directions. One is the tangent, along the velocity. The other is the normal, at right angles to the tangent. Split the acceleration \(\vec a\) along these two directions. Motion in Two Dimensions called the parts \(a_{\parallel}\) and \(a_{\perp}\). On a curved path they have their own names.
The part along the tangent is the tangential acceleration, written \(a_t\). It is the rate of change of the speed, \(a_t = dv/dt\). It is positive when the body speeds up and negative when it slows down. The part along the normal is the normal acceleration, written \(a_{\perp}\). It turns the velocity without changing the speed. It always points towards the inside of the bend.
The angle between \(\vec a\) and \(\vec v\) tells you what happens to the speed. If the angle is less than \(90°\), \(a_t\) is positive and the body speeds up while it turns. If the angle is more than \(90°\), \(a_t\) is negative and the body slows down while it turns. If the angle is exactly \(90°\), \(a_t = 0\). The body then turns at a steady speed.
The S-bend shows two more facts. On a straight road the normal acceleration is zero, however fast the car goes. On a bend the normal acceleration is never zero, even when the speed is steady. Its size depends on the speed and on how sharply the road bends. Lesson 3 finds that size for a circle, and Lesson 6 for any curve.
An auto-rickshaw on a bend has an acceleration of \(2.5\,\text{m/s}^2\), directed at \(53°\) to its velocity. Take \(\cos 53° = 0.6\) and \(\sin 53° = 0.8\). Find the tangential and the normal acceleration. Is the auto speeding up or slowing down?
The tangential part lies along the velocity: \(a_t = a\cos 53° = 2.5 \times 0.6 = 1.5\,\text{m/s}^2\).
The normal part lies across the velocity: \(a_{\perp} = a\sin 53° = 2.5 \times 0.8 = 2.0\,\text{m/s}^2\).
Check: \(\sqrt{1.5^2 + 2.0^2} = \sqrt{6.25} = 2.5\,\text{m/s}^2\). The angle is less than \(90°\), so \(a_t\) is positive and the auto is speeding up while it turns.
A body's acceleration is \(5\,\text{m/s}^2\), at \(37°\) to its velocity. Take \(\sin 37° = 0.6\) and \(\cos 37° = 0.8\). What is the size of its normal acceleration, in m/s²?
At one instant, a body's acceleration makes an angle of \(120°\) with its velocity. Which statements are true at that instant?
The simplest curve
Go back to the fourth chip of the first figure. There the acceleration was turned at every instant so that it stayed across the velocity. So \(a_t\) was zero all the time, and the speed never changed. The size of the acceleration was also kept the same. The velocity then turned by the same angle in every second, and the path closed into a circle.
This is why circular motion comes first among curved motions. On a circle, the direction of the tangent turns steadily, and every point of the path is the same distance from one centre. A car on a roundabout, a stone whirled on a string and a point on a turning fan all move on circles. The rest of this chapter describes such motion with angles, and then extends the ideas to any curve.
The acceleration of a body on a curved path does not point along the path. Many students draw it along the velocity. Only the tangential part lies along the velocity. On any bend there is also a normal part, and on a steady bend that part is the whole acceleration.
A body's acceleration stays at right angles to its velocity at every instant. What happens to its speed?
Gravity acts on a dropped ball and on a thrown ball in exactly the same way. Explain why one path is straight and the other is curved.
The shape of a path depends on the angle between the velocity and the acceleration, not on the acceleration alone. The dropped ball's velocity is vertical, along gravity, so gravity only changes its speed and the path stays straight. The thrown ball's velocity makes an angle with the vertical. Gravity then has a part across the velocity, which turns the velocity downward at every instant, and the path bends.
- Thinking that a body under a constant acceleration must move in a straight line.A constant acceleration at an angle to the velocity bends the path, as for a projectile. The path is straight only when the acceleration lies along the velocity.
- Drawing the acceleration of a body on a bend along the path.On a bend there is always a normal part, pointing to the inside of the bend. Only \(a_t\) lies along the path.
- Saying that a car on a bend at steady speed has no acceleration.Steady speed means \(a_t = 0\). The normal acceleration \(a_{\perp}\) is still there, because the direction of the velocity is changing.