Kinematics: displacement, velocity and acceleration Problem set B One question at a time

A bead slides along a straight wire. The displacement from a fixed point is \(s\) cm, where \(s=3 t^{3} - 10 t^{2} + 7 t + 2\) and \(t\ge0\) is in seconds.

1. (a) Find the velocity when \(t=2\).

   • \(v=\frac{ds}{dt}=9 t^{2} - 20 t + 7\)
   • \(v(2)=3\)

2. (b) Find the acceleration when \(t=2\).

   • \(a=\frac{dv}{dt}=18 t - 20\)
   • \(a(2)=16\)

A raft drifts along a straight river. The displacement from a fixed point is \(s=20e^{-0.5t}+2t\) m, where \(t\ge0\) is in seconds. Give your answers to 3 significant figures.

1. (a) Find the velocity when \(t=1\).

   • \(v=2-10e^{-0.5t}\)
   • \(v(1)=−4.07\)

2. (b) Find the acceleration when \(t=1\).

   • \(a=5e^{-0.5t}\)
   • \(a(1)=3.03\)

A sled slides along a straight slope. The displacement from a fixed point is \(s=2 t^{3} - 21 t^{2} + 60 t\) m, for \(t\ge0\) seconds.

1. (a) Find the times when the particle is at rest.

   • \(v=6 t^{2} - 42 t + 60\)
   • \(v=0\Rightarrow 6 \left(t - 5\right) \left(t - 2\right)=0\)
   • \(t=2,\ t=5\)

2. (b) Find the displacement at each of those times.

   • \(s(2)=52\)
   • \(s(5)=25\)

A weight bobs on an elastic string. The displacement from its rest position is \(x=5\cos(2t)\) cm, where \(t\) is in seconds. Use radians.

1. (a) Find the velocity when \(t=1\). Give your answer to 3 significant figures.

   • \(v=\frac{dx}{dt}=- 10 \sin{\left(2 t \right)}\)
   • \(v(1)=−9.09\)

2. (b) Find the first time after \(t=0\) when the object is at rest.

   • \(v=0\Rightarrow \sin(2t)=0\Rightarrow 2t=\pi\)
   • \(t=1.57\)

3. (c) Write down the greatest speed of the object.

   • \(|v|\le 10\)
   • The greatest speed is 10 cm s⁻¹.

A cyclist rides along a straight road. Its displacement from O is \(s=t^{3} - 15 t^{2} + 63 t\) metres after \(t\) seconds, \(t\ge0\). For each time, say whether the particle is speeding up or slowing down.

1. (a) \(t=2\)

   • \(v=3 t^{2} - 30 t + 63,\ a=6 t - 30\)
   • \(v(2)=15,\ a(2)=−18\)
   • v and a have opposite signs, so the particle is slowing down.

2. (b) \(t=4\)

   • \(v=3 t^{2} - 30 t + 63,\ a=6 t - 30\)
   • \(v(4)=−9,\ a(4)=−6\)
   • v and a have the same sign, so the particle is speeding up.

3. (c) \(t=6\)

   • \(v=3 t^{2} - 30 t + 63,\ a=6 t - 30\)
   • \(v(6)=−9,\ a(6)=6\)
   • v and a have opposite signs, so the particle is slowing down.

A skater glides along a straight line of ice. Its acceleration is \(a=6 t - 10\) m s\(^{-2}\). When \(t=0\) the velocity is 7 m s\(^{-1}\) and the displacement is -1 m.

1. (a) Find an expression for the velocity.

   • \(v=\int a\,dt=3 t^{2} - 10 t+c\)
   • \(v(0)=7\Rightarrow c=7\)

2. (b) Find an expression for the displacement.

   • \(s=\int v\,dt=t^{3} - 5 t^{2} + 7 t+k\)
   • \(s(0)=-1\Rightarrow k=-1\)

3. (c) Find the displacement when \(t=3\).

   • \(s(3)=2\)

A sledge slides along a flat snowfield after a push. Its distance from the start is \(s=45\left(1-e^{-0.15t}\right)\) metres after \(t\) seconds, \(t\ge0\).

1. (a) Find the velocity when \(t=4\). Give your answer to 3 significant figures.

   • \(v=6.75e^{-0.15t}\)
   • \(v(4)=3.70\)

2. (b) Find the time when the velocity is 0.5 m s\(^{-1}\). Give your answer to 3 significant figures.

   • \(6.75e^{-0.15t}=0.5\)
   • \(t=-\frac{1}{0.15}\ln\left(\frac{0.5}{6.75}\right)=17.4\)

3. (c) Find the distance that the sledge approaches and never passes.

   • \(t\to\infty:\ e^{-0.15t}\to0,\ s\to45\)

The top of an engine piston moves up and down. Its height above the ground is \(h=15-6\cos(1.2t)\) cm after \(t\) seconds, where \(t\ge0\). Use radians.

1. (a) Find the vertical velocity when \(t=2\). Give your answer to 3 significant figures.

   • \(v=\frac{dh}{dt}=7.2\sin(1.2t)\)
   • \(v(2)=4.86\)

2. (b) Find the greatest rate at which the piston rises or falls.

   • The sine function has greatest value 1.
   • The greatest rate is 7.2 cm s⁻¹.

3. (c) Find the first time that the piston is rising at 4 cm s\(^{-1}\). Give your answer to 3 significant figures.

   • \(7.2\sin(1.2t)=4\)
   • Solve the equation with the calculator (radian mode), or by taking the inverse sine.
   • The first time is t = 0.491 s.

A train moves along a straight section of line. Its velocity is \(v=t^{2} - 8 t + 12\) m s\(^{-1}\) for \(0\le t\le7\). Give your answers to 3 significant figures.

1. (a) Find the times when the velocity is zero.

   • \(t^{2} - 8 t + 12=0\)
   • \(t=2,\ t=6\)

2. (b) Find the displacement from the starting point after 7 seconds.

   • \(\int_{0}^{7}v\,dt=2.33\)
   • Forwards and backwards parts partly cancel.

3. (c) Find the total distance travelled in the 7 seconds.

   • \(\int_{0}^{7}|v|\,dt=23.7\)
   • The distance is 23.7 m.

A probe moves along a straight tube. Its displacement from O is \(s=8te^{-0.5t}\) metres after \(t\) seconds, \(t\ge0\). Give your answers to 3 significant figures where needed.

1. (a) Find expressions for the velocity and the acceleration.

   • Product rule with \(u=8t\) and the exponential factor.
   • \(v=8(1-0.5t)e^{-0.5t}\)
   • \(a=4(0.5t-2)e^{-0.5t}\)

2. (b) Find the time when the particle is at rest, and its displacement then.

   • \(v=0\Rightarrow 1-0.5t=0\Rightarrow t=2\)
   • \(s(2)=5.89\)

3. (c) Find the time when the acceleration is zero, and the velocity then.

   • \(a=0\Rightarrow 0.5t-2=0\Rightarrow t=4\)
   • \(v(4)=−1.08\)
   • The velocity is negative: the particle is moving back towards O.

A drone moves along a straight line. Its velocity is \(v=6\sin(0.5t)-2.5\) m s\(^{-1}\) for \(0\le t\le9\). Use radians and give your answers to 3 significant figures.

1. (a) Find the times when the particle is at rest.

   • \(6\sin(0.5t)=2.5\)
   • \(t=0.860\text{ and }t=5.42\)
   • Use the calculator to find both solutions in the interval.

2. (b) Find the total distance travelled in the 9 seconds.

   • \(\int_{0}^{9}|v|\,dt=28.8\)
   • The velocity changes sign at both times, so use the absolute value.

3. (c) Find the displacement from the starting point after 9 seconds.

   • \(\int_{0}^{9}v\,dt=−7.97\)