Equations of motion
1 The displacement, s in metres, of an object after a time, t in seconds, is given by s = 90t – 4 t2
(a) Find by differentiation the equation for its velocity.
(b) At what time will the velocity be zero?
(c) Show that the acceleration is a constant and state its value.
2 For an object moving with constant acceleration, show by integration that the velocity, v, is given by v = u + at.
State clearly the meaning of the symbol, u, in this equation.
- For a body moving with velocity v = u + at, show by integration that s = ut + ½ at2.
Where the symbols have their usual meaning.
- The displacement, s, of a moving object after a time, t, is given by s = 8 – 10t + t2.
Show that the unbalanced force acting on the object is constant.
- The displacement, s, of an object after time, t, is given by s = 3t3 + 5t.
(a) Derive an expression for the acceleration of the object.
(b) Explain why this expression indicates that the acceleration is not constant.
- A trolley is released from the top of a runway which is 6 m long.
The displacement, s in metres, of the trolley is given by the expression s = 5t + t2, where t is in seconds.
Determine:
(a) an expression for the velocity of the trolley
(b) the acceleration of the trolley
(c) the time it takes the trolley to reach the bottom of the runway
(d) the velocity of the trolley at the bottom of the runway.
- A box slides down a smooth slope with an acceleration of 4 m s-2. The velocity of the box at a time t = 0 is 3 m s-1 down the slope.
Show by integration that the velocity, v, of the box is given by v = 3 + 4t.
- The equation for the velocity, v, of a moving trolley is v = 2 + 6t.
Derive an expression for the displacement, s, of the trolley.
9. A projectile is launched from the top of a building with an initial speed of 20 m s-1 at an angle of 30° to the horizontal. The height of the building is 30 m.
(a) Calculate how long it takes the projectile to reach the ground.
(b) Calculate the velocity of the projectile on impact with the ground, (magnitude and direction).
