M2 questions

(These are from review exercise 1 in the Heinemann book.)

5. The unit vectors i and j are horizontal and vertically upwards respectively. A particle is projected with velocity (8i+10j)m/s from a point O at the top of a cliff and moves freely under gravity. Six seconds after projection, the particle strikes the sea at the point S. Calculate:
a) the horiz distance between O and S (-done; answer: 48m)
b) the vert distance between O and S (-done; answer: 116m)
At time T seconds after projection, the particle is moving with velocity (8i-14.5j)m/s.
c) Find the value of T and the position vector, relative to O, of the particle at this instant.

I'm having trouble with part c. I got T=2.5, which is correct, and position vector 20i-36.25j, but the correct answer is 20i-5.625j, and I have no idea how to get it! :redface:

35. A ball is projected from a point A on horiz ground, with speed 14m/s at an angle of elevation @, and moves freely under gravity. At the instant when the ball is at point P, which is 5m above the ground, the components of velocity of the ball horiz & vertically upwards are both u m/s.
a) By using energy considerations, or otherwise, show that u=7. (-done)
b) Obtain the value of @. (-done; answer: 60 degrees)
c) Find, in seconds to 2 decimal places, the time taken by the ball to reach its greatest height above the ground. (answer: 1.24s)
The ball continues to move until it strikes, at right angles, a fixed plane inclined at an angle 50 to the horizontal.
d) Calculate, to 1 decimal place, the value of V, the speed of the ball when it strikes the plane. (answer: 9.1 m/s)

I did parts a & b. I think my method for part c is correct (it worked in all the other questions), but my answer is slightly off. I have no idea how to do part d.

40. The non-gravitaional resistive forces opposing the motion of a lorry of mass 2200kg are constant and total 3100N.
a) The lorry is moving along a straight horiz road at a constant speed of 20m/s. Calculate, in kW, the rate at which the engine of the lorry is working. (-done; answer: 62kW)
b) If this rate of working is suddenly decreased by 12kW, find, in m/s², the immediate retardation of the lorry. (answer: 3/11 m/s²)

I've tried doing part b, but I can't get the same answer...

Thanks. :smile:

Reply 1

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(These are from review exercise 1 in the Heinemann book.)

5. The unit vectors i and j are horizontal and vertically upwards respectively. A particle is projected with velocity (8i+10j)m/s from a point O at the top of a cliff and moves freely under gravity. Six seconds after projection, the particle strikes the sea at the point S. Calculate:
a) the horiz distance between O and S (-done; answer: 48m)
b) the vert distance between O and S (-done; answer: 116m)
At time T seconds after projection, the particle is moving with velocity (8i-14.5j)m/s.
c) Find the value of T and the position vector, relative to O, of the particle at this instant.

I'm having trouble with part c. I got T=2.5, which is correct, and position vector 20i-36.25j, but the correct answer is 20i-5.625j, and I have no idea how to get it! :

You've got t=2.5, and the horizontal component of the position vector = 20.
The vertical component comes from putting t=2.5 into

s = ut + 0.5a(t^2)
s= 10*2.5 -4.9*(25/4)
s= -5.625 (i.e. below 0)

OK?
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Reply 2

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(These are from review exercise 1 in the Heinemann book.)

35. A ball is projected from a point A on horiz ground, with speed 14m/s at an angle of elevation @, and moves freely under gravity. At the instant when the ball is at point P, which is 5m above the ground, the components of velocity of the ball horiz & vertically upwards are both u m/s.
a) By using energy considerations, or otherwise, show that u=7. (-done)
b) Obtain the value of @. (-done; answer: 60 degrees)
c) Find, in seconds to 2 decimal places, the time taken by the ball to reach its greatest height above the ground. (answer: 1.24s)
The ball continues to move until it strikes, at right angles, a fixed plane inclined at an angle 50 to the horizontal.
d) Calculate, to 1 decimal place, the value of V, the speed of the ball when it strikes the plane. (answer: 9.1 m/s)

I did parts a & b. I think my method for part c is correct (it worked in all the other questions), but my answer is slightly off. I have no idea how to do part d.
:smile:

I get 1.2372 for c.

Draw a sketch for d.
If the plane is at 50 deg to horizontal, the ball strikes it at 40 deg to the horizontal. Sketch should show this.

path is at 40 deg to horizontal when vertcomponent/horizcomponent = tan40
you have horizcomponent =7, so vertcomponent = 7*tan 40 = 5.87

speed is sqrt(vertcomponent^2+horizcomponent^2) = 9.137

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Reply 3

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(These are from review exercise 1 in the Heinemann book.)
40. The non-gravitaional resistive forces opposing the motion of a lorry of mass 2200kg are constant and total 3100N.
a) The lorry is moving along a straight horiz road at a constant speed of 20m/s. Calculate, in kW, the rate at which the engine of the lorry is working. (-done; answer: 62kW)
b) If this rate of working is suddenly decreased by 12kW, find, in m/s², the immediate retardation of the lorry. (answer: 3/11 m/s²)

I've tried doing part b, but I can't get the same answer...
Thanks. :smile:

P = 50000= 20F => F=2500 (RESULTANT FORCE = 2500-3100 = -600)

RF = ma
-600 = 2200a
a=-600/2200

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Reply 4

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OP

Thanks!

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